Available online at ScienceDirect. Procedia Computer Science 74 (2015 ) 47 52


 Albert Atkins
 1 years ago
 Views:
Transcription
1 Available online at ScienceDirect Procedia Computer Science 74 (05 ) 47 5 International Conference on Graph Theory and Information Security Fractional Metric Dimension of Tree and Unicyclic Graph Daniel A. Krismanto, Suhadi Wido Saputro Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesa 0, Bandung 403, Indonesia Abstract A vertex v in a simple connected graph G resolves two vertices x and y in G if the distance from x to v is not equal to distance from y to v. The vertex set R{x, y} is defined as the set of vertices in G which resolve x and y. A function f : V(G) [0, ] is called a resolving function of G if f (R{x, y}) for any two distinct vertices x and y in G. The minimal value of f (V(G)) for all resolving functions f of G is called the fractional metric dimension of G. In this paper, we determine the fractional metric dimension of G where G is a tree or G is a unicyclic graph. c 05 The Authors. Published by Elsevier by Elsevier B.V. B.V. This is an open access article under the CC BYNCND license Peerreview ( under responsibility of the Organizing Committee of ICGTIS 05. Peerreview under responsibility of the Organizing Committee of ICGTIS 05 Keywords: Fractional metric dimension, resolving function, tree graph, unicyclic graph. 00 MSC: 05C, 05C78. Introduction Throughout this paper, all graphs G are finite, connected, and simple. We denote by V the vertex set of G and by E the edge set of G. The distance between two vertices u, v V(G), denoted by d(u, v), is the length of a shortest path from u to v in G. Foru, v V(G), we define R{u, v} = {z V(G) d(u, z) d(v, z)}. A vertex set W V(G) is called a resolving set of G if W R{u, v} for any two distinct vertices u, v V(G). The minimum cardinality of all resolving sets of G is called the metric dimension of G. The metric dimension concept has been applied to many various areas including coin weighing problem [8], robot navigation [7], and strategies for the mastermind game [3]. Determining the metric dimension of a graph was formulated as an integer programming problem by Chartrand et al. []. Furthermore, Currie and Oellermann [4] defined fractional metric dimension as the optimal solution of the linear relaxation of the integer programming problem. Let f be a function assigning each vertex v of G a real number f (u) [0, ]. For W V(G), denote f (W) = v W f (v). The function f is called a resolving function of G if f (R{u, v}) for any two distinct vertices u and v in G. The minimum value of f (V(G)) for any resolving function f of G is called the fractional metric dimension of G, denoted by dim f (G). address: The Authors. Published by Elsevier B.V. This is an open access article under the CC BYNCND license ( Peerreview under responsibility of the Organizing Committee of ICGTIS 05 doi:0.06/j.procs
2 48 Daniel A. Krismanto and Suhadi Wido Saputro / Procedia Computer Science 74 ( 05 ) 47 5 The fractional metric dimension problem was initiated by Arumugam and Mathew [] in 0. They provided asufficient condition for a connected graph G whose fractional metric dimension is V(G). They also determined dim f (G) where G is Petersen graph, cycles, hypercubes, stars, wheels, friendship graph, and grids. Some authors applied this topic to some product graphs. Feng et al. [5] have determined the fractional metric dimension of Cartesian product of two graphs. In other paper [6], Feng et al. also studied the fractional metric dimension of corona and lexicographic product graphs. In this paper we consider tree and unicyclic graph. Unicyclic graph is a graph obtained from a tree by adding one more edge. Note that unicyclic graph contains one cycle. The study of fractional metric dimension for a tree has been initiated by Arumugam and Mathew []. They have proved that dim f (P n ) =. They also provided an exact value for the fractional metric dimenison of star and bistar. In this paper, we are interested to determine the fractional metric dimension of any trees. We also investigate the fractional metric dimension of unicyclic graph.. Fractional metric dimension of tree Let G be a tree and v be a vertex of G. Abranch of G at v is defined as a maximal subtree of G containing v as an end point. So, if degree of v is k, then v has at most k different branches. A branch of v which is isomorphic to a path is called a leaf of v. If there is a leaf from v, then v is called a stem of G. A stem vertex v is called a node if v has more than one leaves. Those definitions are firstly introduced by Slater [9]. For i n and m i, we define a generalized star graph S n (m, m,...,m n ) as a tree graph having m + m m n + vertices and containing a vertex of degree n which has n leaves where the length of the ith leaf is m i. Note that if v is a node in a tree graph G, then an induced subgraph of G by v and all of its leaves is isomorphic to a generalized star graph. Lemma. Let G be a generalized star graph and w be a node in G. Let u V(G) where degree of u is. If two distinct vertices x, y V(G) satisfy w R{x, y}, then u R{x, y}. Proof. Suppose that x and y are two distinct vertices in G such that w R{x, y} but u R{x, y}. Let L and L are leaves of w which are containing x and y respectively. Since u R{x, y}, wehaved(x, u) = d(y, u). So, u V(L ) and u V(L ). Therefore, we obtain d(x, u) = d(x, w) + d(w, u) and d(y, u) = d(y, w) + d(w, u). It follows that d(x, w) = d(y, w) which implies w R{x, y}, a contradiction. Lemma. For n 3, let G be a generalized star graph S n (m, m,, m n ). Then dim f (G) = n. Proof. First, we will prove that dim f (G) n. We define the function f : V(G) [0, ] where f (v) =, if degree of v is, For any two different vertices u, v V(G) and also considering Lemma, R{u, v} contains two distinct vertices of degree. So, we have f (R{u, v}). Therefore, f is a resolving function of G. Hence, dim f (G) f (V(G)) = n. Now, we will prove dim f (G) n. Let g be a minimum resolving function of G. Let x and y be two different vertices in G. Then there exist two different vertices u, v V(G) from two different leaves of G such that u, v R{x, y}. Let L u and L v be two leaves of G containing u and v respectively. Note that V(L u ) V(L v ) R{x, y}. So, g(v(l u ) V(L v )). It follows that for every leaf L of G,wehaveg(V(L)). Hence, dim f (G) n. Using both Lemmas and, we can obtain the following theorem. Theorem 3. Let G be a tree graph with k nodes w, w,...,w k. Let L(w) be the number of leaf of node w of G. Then dim f (G) = k L(w i ) i=
3 Daniel A. Krismanto and Suhadi Wido Saputro / Procedia Computer Science 74 ( 05 ) Proof. To proof Theorem 3 above, we use the following definitions. For i k, we define S i as an induced subgraph of G by w i and all vertices of its leaves. Note that S i is isomorphic to a generalized star graph. Let S = V(S ) V(S )... V(S k ). First, we define a function f : V(G) [0, ] as f (v) =, if v S and degree of v is, For any two different vertices u, v V(G), R{u, v} contains two distinct vertices of S of degree. So, we have f (R{u, v}). Therefore, f is a resolving function of G. Hence, dim f (G) f (V(G)) = ki= Now, let g be a minimal resolving function of G. Since S i is a generalized star graph for i k, by Lemma we obtain dim f (G) g(s ) = ki= Remark 4. Theorem 3 above strengthen Arumugam and Mathew s result of star graph. 3. Fractional metric dimension of unicyclic graph In this section, let G be a unicyclic graph. Let R be an induced subgraph of G which is isomorphic to a cycle. A vertex v V(R) is called a root if degree of v is more than. If a root v has degree 3 and G v has a path component, then v is called a grass root. We also consider a vertex set Q V(R) which is a set of all vertices of degree at least 3 in R and vertex set A Q which is a maximal subset of Q such that for every u, v A, d(u, v). For vertex v V(R), we still use the definitions of branch, leaf, stem, and node as stated in Section. Let A, B, C, D, E be collections of unicyclic graphs G where:. A is a set of G which is isomorphic to a cycle graph.. B is a set of G having only grass root(s). Furthermore, we decompose B into 5 partitions B, B,...,B 5 where: (a) G B if G Band R of G is isomorphic to C 4. (b) G B if G Band R of G is an even cycle of order at least 6 and A 3. (c) G B 3 if G Band R of G is an even cycle of order at least 6 and A 4. (d) G B 4 if G Band R is isomorphic to C 3. (e) G B 5 if G Band R of G is an odd cycle of order at least C is a set of G having exactly one root which is not grass root. 4. D is a set of G having exactly two roots which are not grass root. Furthermore, we decompose D into partitions D and D where: (a) G D if G D, R is isomorphic to even cycle, and for two distinct roots u, v V(G), d(u, v) =. (b) G D if G Dbut G D. 5. E is the set of G which is not in class A, B, C,orD. In theorem below, we give an exact values of fractional metric dimension of a unicyclic graph G in some collections. Theorem 5. Let G be a unicyclic graph of order n 3 having k nodes w, w,...,w k. Let L(w) be the number of leaf of node w of G. Let R be an induced subgraph of G which is isomorphic to a cycle. Let Q V(R) be a set of all vertices of degree at least 3 in R and A Q be a maximal subset of Q such that for every u, v A, d(u, v).if G B 5, then, if G Aand R is even cycle,, if G Aand R is odd cycle,, if G B, 3 dim f (G) =, if G B B 4, A A +, if G B 3, + ki= L(w i ), if G C D, ki= L(w i ), if G D E.
4 50 Daniel A. Krismanto and Suhadi Wido Saputro / Procedia Computer Science 74 ( 05 ) 47 5 Proof. Note that all unicyclic graphs in A are isomorphic to cycle graph. The fractional metric dimension of cycle graph has been determined by Arumugam and Mathew []. Now, we assume that G A B 5. Case 5.. G B First, we define a function f : V(G) [0, ] such that for v V(G), we have, if degree of v is, f (v) =, if degree of v is and v V(R), Let X be vertex set of G containing all vertices of degree. Let X be vertex set of R containing all vertices of degree. Note that for any two different vertices u, v V(G), there exist two distinct vertices x, y X X such that x, y R{u, v}. So, we have f (R{u, v}). Therefore, f is a resolving function of G. Hence, dim f (G) f (V(G)) =. Now, let g be a minimal resolving function of G. Note that there exist two different pair of vertices (a, b) and (c, d) in V(G) such that R{a, b} R{c, d} =. It follows that dim f (G) g (R{a, b} R{c, d}). Case 5.. G B B 3 We define vertex set X as the set of all vertices of R where for every x X there exists a A such that d(x, a) =. If A X 6, we define vertex set C V(R) \ (A X) such that C = 6 A X and for every x C there exists y C such that d(x, y) =. We also define B as the subset of X where degree of every vertex of B is. Note that A, B, and C are disjoint. Let V be a subset of V(G) containing all vertices of degree, and S = V B C. First, we define a function f : V(G) [0, ] such that for v V(G), we have 4, if v S and G B, f (v) = A +, if v S and G B 3, Note that for any two different vertices u, v V(G), if G B, then R{u, v} S 4, otherwise R{u, v} S A +. So, we have f (R{u, v}). Therefore, f is a resolving function of G. Hence, dim f (G) f (V(G)) = 3 for G B and dim f (G) f (V(G)) = A A + for G B 3. For i A, let w i A, x i and y i be two different vertices in V(R) such that d(x i, w i ) = d(y i, w i ) =, and z i V(R) such that d(w i, z i ) =. Let H i = R{x i, z i } R{y i, z i }. We distinguish two cases.. G B Let g be a minimal resolving function of G. Since V(G) = (R{x i, z i } R{y i, z i })\H i, we obtain g (V(G))+g (H i ). Since R{x i, y i } V(G) \ H i, we obtain g (V(G)) g (H i ). Therefore, we have dim f (G) g (V(G)) 3.. G B 3 Suppose that g be a minimal resolving function of G where g (V(G)) < A A +. Since V(G) = (R{x i, z i } R{y i, z i }) \ H i, we obtain g (V(G)) + g (H i ). Since R{x i, y i } V(G)\ H i, we obtain g (H i ) V(G) = V(H ) V(H )... V(H A ). Therefore, A g (V(G)) = g (H i ) A A +. So, it is impossible to have a minimal resolving function g of G where g (V(G)) < A A +. i= A +. Note that Case 5.3. G B 4 Let S be a vertex set of G containing all vertices of G of degree and all vertices of R of degree. First, we define a function f 3 : V(G) [0, ] such that for v V(G), we have f 3 (v) = Note that for any two different vertices u, v V(G), R{u, v} S. So, we have f 3 (R{u, v}). Therefore, f 3 is a resolving function of G. Hence, dim f (G) f 3 (V(G)) = 3.
5 Daniel A. Krismanto and Suhadi Wido Saputro / Procedia Computer Science 74 ( 05 ) Now, let g 3 be a minimal resolving function of G. Let V(R) = {x, x, x 3 } and L i be a leaf of x i for i 3. Note that for distinct i, j {,, 3}, V(L i ) V(L j ) R{x i, x j }. Without loss of generality, let g 3 (V(L )) = t for t [0, ]. So, we obtain g 3 (V(L )) = t and g 3 (V(L 3 )) = max{t, t}. It follows that t =. So, we have that dim f (G) g 3 (V(G)) = 3. Case 5.4. G C D For i k, we define a vertex set S i as the set of all vertices of degree one in leaf of node w i. Choose a pair of vertices x, y V(R) such that d(x, y) = and for a {x, y}, either degree of a is or a is a grass root of G. We define S = S S... S k and T = S {x, y}. First, we define a function f 4 : V(G) [0, ] such that for v V(G), we have f 4 (v) =, if v T, Note that for any two different vertices u, v V(G), R{u, v} T. So, we have f 4 (R{u, v}). Therefore, f 4 is a resolving function of G. Hence, dim f (G) f 4 (V(G)) = + ki= For the lower bound, let g 4 be a minimal resolving function of G. For i k, let F i be an induced subgraphs of G by S i. Note that F i is a generalized star graph. By Lemma, we obtain dim f (F i ) = L(w i). So, we can say that dim f (S ) = ki= However, we also always can find a pair of vertices u, v V(G) such that R{u, v} S =. Therefore, we obtain that dim f (G) g 4 (V(G)) + ki= Case 5.5. G D E For i k, we define a vertex set S i as the set of all vertices of degree one in leaf of node w i. We define S = S S... S k. First, we define a function f 5 : V(G) [0, ] such that for v V(G), we have f 5 (v) = Note that for any two different vertices u, v V(G), R{u, v} S. So, we have f 5 (R{u, v}). Therefore, f 5 is a resolving function of G. Hence, dim f (G) f 5 (V(G)) = ki= For the lower bound, let g 5 be a minimal resolving function of G. For i k, let F i be an induced subgraphs of G by S i. Note that F i is a generalized star graph. By Lemma, we obtain dim f (F i ) = L(w i). So, we can say that dim f (S ) = ki= Therefore, we obtain that dim f (G) g 5 (V(G)) ki= In Theorem 6, we give the upper bound of dim f (G) for G B 5. We also show that the bound is sharp. In Theorem 7, we provide an existence of unicyclic graph G B 5 such that dim f (G) is not equal to the upper bound in Theorem 6. Theorem 6. Let G be a unicyclic graph in class B 5. Let R be an induced subgraph of G which is isomorphic to cycle graph. Then dim f (G) + 3. The bounds is sharp. Proof. Let S = {v V(G) degree of v is } {v V(R) degree of v is }. We define a function f : V(G) [0, ] such that for v V(G), we have { f (v) = +3 Note that for any two different vertices u, v V(G), R{u, v} S +3. So, we have f (R{u, v}). Therefore, f is a resolving function of G. Hence, dim f (G) f (V(G)) = +3. Now, we consider a unicyclic G which is obtained from C n where n 5 is odd and every vertex of C n has one leaf. We will prove that dim f (G) = +3. Now, we need to show that dim f (G) +3.
6 5 Daniel A. Krismanto and Suhadi Wido Saputro / Procedia Computer Science 74 ( 05 ) 47 5 Let g be a resolving function of G. Let V(R) = {x 0, x,...,x n } where R = x 0 x...x n x 0. For 0 i n, let y i V(R) buty i x i E(G) and L(x i ) be a leaf of x i. Note that R{x i mod n, y (i ) mod n } = V(G) \ (V(L(x ( n+ +i) mod n))... V(L(x (n +i) mod n )) {x i }). Since g(r{a, b}) for every two distinct vertices a, b V(G), we obtain that n g(r{x i mod n, y (i ) mod n }) = n g(v(g)) n 3 i=0 Since g(x i ) 0 for i n, we will obtain that g(v(g)) n n+3. n g(v(g)) g(x i ) n Theorem 7. There exists a unicyclic graph G B 5 such that for an induced subgraph R of G which is isomorphic to cycle graph, dim f (G) < +3. Proof. Let G be a unicyclic graph having an odd cycle R with 7 and 3 grass roots such that grass roots a, b V(G) are adjacent and another grass root c V(G) satisfies d(c, a) = d(c, b) =. We will prove that dim f (G) = 5 4. Note that for 7, we have 5 5( + 3) = 4 4( + 3) < 8 4( + 3) = + 3. First, let S = {u V(G) degree of u is } {v V(R) vc E(G)}. Now, we define a function f : V(G) [0, ] such that for v V(G), we have f (v) = 4 Note that for any two different vertices u, v V(G), R{u, v} S 4. So, we have f (R{u, v}). Therefore, f is a resolving function of G. Hence, dim f (G) f (V(G)) = 5 4. For the lower bound, suppose that dim f (G) < 5 4. Let g be a minimal resolving function of G such that g(v(g)) < 5 4. Let y a, y b, y c V(R) such that for t {a, b, c}, ty t E(G). Let v, v V(R) be two distinct vertices which are adjacent to c and L w be a leaf of node w in R. Since R{v, v } = V(G) \ V(L c ), we obtain g(v(l c )) = 4 k for positive k R. Now, we consider R{y a, b} and R{y b, a}. We can say that R{y a, b} = V(L c ) X and R{y b, a} = V(L c ) Y. So, we obtain that g(x) = k = g(y). Note that X Y = and R{v, v } = V(G) \ V(L c ) = X Y. It follows that g(v(g)) = g(v(l c )) + g(x) + g(y) = k, a contradiction. i=0 Acknowledgement. This work is supported by Institut Teknologi Bandung, Indonesia by research grant Riset dan Inovasi ITB 05. The authors also are very grateful to the referees for their careful reading with corrections and useful comments. References. Arumugam S, Mathew V. The fractional metric dimension of graphs. Discrete Math. 0;3: Chartrand G, Eroh L, Johnson MA, Oellermann OR. Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 000;05: Chvátal V. Mastermind. Combinatorica 983;3: Currie J, Oellermann OR. The metric dimension and metric independence of a graph. J. Combin. Math. Combin. Comput. 00;39: Feng M, Lv B, Wang K. On the fractional metric dimension of graphs. Discrete Appl. Math. 04;70: Feng M, Wang K. On the fractional metric dimension of corona product graphs and lexicographic product graphs. arxiv:06.906v. 7. Khuller S, Raghavachari B, Rosenfeld A. Landmarks in graphs. Discrete Appl. Math. 996;70: Shapiro H, Soderberg S. A combinatory detection problem. Amer. Math. Monthly 963;70: Slater PJ. Leaves of trees. Congr. Numer. 975;4:
The Metric Dimension of Graph with Pendant Edges
The Metric Dimension of Graph with Pendant Edges H. Iswadi 1,2, E.T. Baskoro 2, R. Simanjuntak 2, A.N.M. Salman 2 2 Combinatorial Mathematics Research Group Faculty of Mathematics and Natural Science,
More informationChapter 2. Basic Terminology and Preliminaries
Chapter 2 Basic Terminology and Preliminaries 6 Chapter 2. Basic Terminology and Preliminaries 7 2.1 Introduction This chapter is intended to provide all the fundamental terminology and notations which
More informationUnicyclic Graphs with Given Number of Cut Vertices and the Maximal Merrifield  Simmons Index
Filomat 28:3 (2014), 451 461 DOI 10.2298/FIL1403451H Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Unicyclic Graphs with Given Number
More informationOdd induced subgraphs in graphs of maximum degree three
Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A longstanding
More informationDiameter of paired domination edgecritical graphs
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 40 (2008), Pages 279 291 Diameter of paired domination edgecritical graphs M. Edwards R.G. Gibson Department of Mathematics and Statistics University of Victoria
More informationGraphs without proper subgraphs of minimum degree 3 and short cycles
Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract
More informationOn the kpath cover problem for cacti
On the kpath cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we
More informationOn the independence number of graphs with maximum degree 3
On the independence number of graphs with maximum degree 3 Iyad A. Kanj Fenghui Zhang Abstract Let G be an undirected graph with maximum degree at most 3 such that G does not contain any of the three graphs
More informationDegree Hypergroupoids Associated with Hypergraphs
Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated
More informationExtremal Graph Theory for Metric Dimension and Diameter
Extremal Graph Theory for Metric Dimension and Diameter Carmen Hernando Departament de Matemàtica Aplicada I Universitat Politècnica de Catalunya Barcelona, Spain carmen.hernando@upc.edu Ignacio M. Pelayo
More informationEvery tree contains a large induced subgraph with all degrees odd
Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University
More informationAll trees contain a large induced subgraph having all degrees 1 (mod k)
All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New
More informationBOUNDARY EDGE DOMINATION IN GRAPHS
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 04874, ISSN (o) 04955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 19704 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA
More informationThe Congestion of ncube Layout on a Rectangular Grid
The Congestion of ncube Layout on a Rectangular Grid S.L. Bezrukov J.D. Chavez L.H. Harper M. Röttger U.P. Schroeder Abstract We consider the problem of embedding the ndimensional cube into a rectangular
More informationGraphical degree sequences and realizations
swap Graphical and realizations Péter L. Erdös Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences MAPCON 12 MPIPKS  Dresden, May 15, 2012 swap Graphical and realizations Péter L. Erdös
More informationOn minimal forbidden subgraphs for the class of EDMgraphs
Also available at http://amcjournaleu ISSN 8966 printed edn, ISSN 897 electronic edn ARS MATHEMATICA CONTEMPORANEA 9 0 6 On minimal forbidden subgraphs for the class of EDMgraphs Gašper Jaklič FMF
More informationThe positive minimum degree game on sparse graphs
The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University
More informationSolutions to Exercises 8
Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.
More informationTHE 0/1BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER
THE 0/1BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER JONATHAN P. MCCAMMOND AND GÜNTER ZIEGLER Abstract. In 1933 Karol Borsuk asked whether every compact subset of R d can be decomposed
More informationExtremal Graphs without ThreeCycles or FourCycles
Extremal Graphs without ThreeCycles or FourCycles David K. Garnick Department of Computer Science Bowdoin College Brunswick, Maine 04011 Y. H. Harris Kwong Department of Mathematics and Computer Science
More informationA 2factor in which each cycle has long length in clawfree graphs
A factor in which each cycle has long length in clawfree graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science
More informationA REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries
Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do
More informationDO NOT REDISTRIBUTE THIS SOLUTION FILE
Professor Kindred Math 04 Graph Theory Homework 7 Solutions April 3, 03 Introduction to Graph Theory, West Section 5. 0, variation of 5, 39 Section 5. 9 Section 5.3 3, 8, 3 Section 7. Problems you should
More informationChromaticchoosability of the power of graphs
Chromaticchoosability of the power of graphs arxiv:109.0888v1 [math.co] 4 Sep 201 SeogJin Kim Young Soo Kwon Boram Park Abstract The kth power G k of a graph G is the graph defined on V(G) such that
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationTools for parsimonious edgecolouring of graphs with maximum degree three. J.L. Fouquet and J.M. Vanherpe. Rapport n o RR201010
Tools for parsimonious edgecolouring of graphs with maximum degree three J.L. Fouquet and J.M. Vanherpe LIFO, Université d Orléans Rapport n o RR201010 Tools for parsimonious edgecolouring of graphs
More informationCycles in a Graph Whose Lengths Differ by One or Two
Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDEBERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS
More informationLarge induced subgraphs with all degrees odd
Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order
More informationAn inequality for the group chromatic number of a graph
Discrete Mathematics 307 (2007) 3076 3080 www.elsevier.com/locate/disc Note An inequality for the group chromatic number of a graph HongJian Lai a, Xiangwen Li b,,1, Gexin Yu c a Department of Mathematics,
More informationUPPER BOUNDS ON THE L(2, 1)LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE
UPPER BOUNDS ON THE L(2, 1)LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE ANDREW LUM ADVISOR: DAVID GUICHARD ABSTRACT. L(2,1)labeling was first defined by Jerrold Griggs [Gr, 1992] as a way to use graphs
More informationThe Goldberg Rao Algorithm for the Maximum Flow Problem
The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }
More informationMIXDECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3. University of Split, Croatia
GLASNIK MATEMATIČKI Vol. 38(58)(2003), 2 232 MIXDECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3 Damir Vukičević University of Split, Croatia
More informationOn a tree graph dened by a set of cycles
Discrete Mathematics 271 (2003) 303 310 www.elsevier.com/locate/disc Note On a tree graph dened by a set of cycles Xueliang Li a,vctor NeumannLara b, Eduardo RiveraCampo c;1 a Center for Combinatorics,
More informationCOMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH. 1. Introduction
COMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the HigmanSims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact
More information1. Relevant standard graph theory
Color identical pairs in 4chromatic graphs Asbjørn Brændeland I argue that, given a 4chromatic graph G and a pair of vertices {u, v} in G, if the color of u equals the color of v in every 4coloring
More informationCharacterizations of Arboricity of Graphs
Characterizations of Arboricity of Graphs Ruth Haas Smith College Northampton, MA USA Abstract The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs
More informationarxiv: v2 [math.co] 30 Nov 2015
PLANAR GRAPH IS ON FIRE PRZEMYSŁAW GORDINOWICZ arxiv:1311.1158v [math.co] 30 Nov 015 Abstract. Let G be any connected graph on n vertices, n. Let k be any positive integer. Suppose that a fire breaks out
More informationGeneralizing the Ramsey Problem through Diameter
Generalizing the Ramsey Problem through Diameter Dhruv Mubayi Submitted: January 8, 001; Accepted: November 13, 001. MR Subject Classifications: 05C1, 05C15, 05C35, 05C55 Abstract Given a graph G and positive
More informationThe 4Choosability of Plane Graphs without 4Cycles*
Journal of Combinatorial Theory, Series B 76, 117126 (1999) Article ID jctb.1998.1893, available online at http:www.idealibrary.com on The 4Choosability of Plane Graphs without 4Cycles* Peter Che Bor
More informationProduct irregularity strength of certain graphs
Also available at http://amc.imfm.si ISSN 18553966 (printed edn.), ISSN 18553974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (014) 3 9 Product irregularity strength of certain graphs Marcin Anholcer
More informationGraph Theory Notes. Vadim Lozin. Institute of Mathematics University of Warwick
Graph Theory Notes Vadim Lozin Institute of Mathematics University of Warwick 1 Introduction A graph G = (V, E) consists of two sets V and E. The elements of V are called the vertices and the elements
More information1 Plane and Planar Graphs. Definition 1 A graph G(V,E) is called plane if
Plane and Planar Graphs Definition A graph G(V,E) is called plane if V is a set of points in the plane; E is a set of curves in the plane such that. every curve contains at most two vertices and these
More informationNonSeparable Detachments of Graphs
Egerváry Research Group on Combinatorial Optimization Technical reports TR200112. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.
More informationCacti with minimum, secondminimum, and thirdminimum Kirchhoff indices
MATHEMATICAL COMMUNICATIONS 47 Math. Commun., Vol. 15, No. 2, pp. 4758 (2010) Cacti with minimum, secondminimum, and thirdminimum Kirchhoff indices Hongzhuan Wang 1, Hongbo Hua 1, and Dongdong Wang
More information3. Eulerian and Hamiltonian Graphs
3. Eulerian and Hamiltonian Graphs There are many games and puzzles which can be analysed by graph theoretic concepts. In fact, the two early discoveries which led to the existence of graphs arose from
More informationHomework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS
Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C
More informationGraphs and Network Flows IE411 Lecture 1
Graphs and Network Flows IE411 Lecture 1 Dr. Ted Ralphs IE411 Lecture 1 1 References for Today s Lecture Required reading Sections 17.1, 19.1 References AMO Chapter 1 and Section 2.1 and 2.2 IE411 Lecture
More informationTree Cover of Graphs
Applied Mathematical Sciences, Vol. 8, 2014, no. 150, 74697473 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ams.2014.49728 Tree Cover of Graphs Rosalio G. Artes, Jr. and Rene D. Dignos Department
More information2.3 Scheduling jobs on identical parallel machines
2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed
More informationThe Independence Number in Graphs of Maximum Degree Three
The Independence Number in Graphs of Maximum Degree Three Jochen Harant 1 Michael A. Henning 2 Dieter Rautenbach 1 and Ingo Schiermeyer 3 1 Institut für Mathematik, TU Ilmenau, Postfach 100565, D98684
More informationMath 443/543 Graph Theory Notes 4: Connector Problems
Math 443/543 Graph Theory Notes 4: Connector Problems David Glickenstein September 19, 2012 1 Trees and the Minimal Connector Problem Here is the problem: Suppose we have a collection of cities which we
More informationON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction
ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove
More informationA simple criterion on degree sequences of graphs
Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree
More information4 Basics of Trees. Petr Hliněný, FI MU Brno 1 FI: MA010: Trees and Forests
4 Basics of Trees Trees, actually acyclic connected simple graphs, are among the simplest graph classes. Despite their simplicity, they still have rich structure and many useful application, such as in
More informationprinceton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora
princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of
More informationConnectivity and cuts
Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every
More informationRational exponents in extremal graph theory
Rational exponents in extremal graph theory Boris Bukh David Conlon Abstract Given a family of graphs H, the extremal number ex(n, H) is the largest m for which there exists a graph with n vertices and
More informationHomework 15 Solutions
PROBLEM ONE (Trees) Homework 15 Solutions 1. Recall the definition of a tree: a tree is a connected, undirected graph which has no cycles. Which of the following definitions are equivalent to this definition
More informationPartitioning edgecoloured complete graphs into monochromatic cycles and paths
arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edgecoloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political
More information8. Matchings and Factors
8. Matchings and Factors Consider the formation of an executive council by the parliament committee. Each committee needs to designate one of its members as an official representative to sit on the council,
More informationOn Total Domination in Graphs
University of Houston  Downtown Senior Project  Fall 2012 On Total Domination in Graphs Author: David Amos Advisor: Dr. Ermelinda DeLaViña Senior Project Committee: Dr. Sergiy Koshkin Dr. Ryan Pepper
More information10. Graph Matrices Incidence Matrix
10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a
More informationDifference Labeling of Some Graph Families
International Journal of Mathematics and Statistics Invention (IJMSI) EISSN: 2321 4767 PISSN: 23214759 Volume 2 Issue 6 June. 2014 PP3743 Difference Labeling of Some Graph Families Dr. K.Vaithilingam
More informationA simple existence criterion for normal spanning trees in infinite graphs
1 A simple existence criterion for normal spanning trees in infinite graphs Reinhard Diestel Halin proved in 1978 that there exists a normal spanning tree in every connected graph G that satisfies the
More informationGraph Isomorphism Completeness for Perfect Graphs and Subclasses of Perfect Graphs
Graph Isomorphism Completeness for Perfect Graphs and Subclasses of Perfect Graphs C. Boucher D. Loker May 2006 Abstract A problem is said to be GIcomplete if it is provably as hard as graph isomorphism;
More informationSome Results on 2Lifts of Graphs
Some Results on Lifts of Graphs Carsten Peterson Advised by: Anup Rao August 8, 014 1 Ramanujan Graphs Let G be a dregular graph. Every dregular graph has d as an eigenvalue (with multiplicity equal
More informationON THE TREE GRAPH OF A CONNECTED GRAPH
Discussiones Mathematicae Graph Theory 28 (2008 ) 501 510 ON THE TREE GRAPH OF A CONNECTED GRAPH Ana Paulina Figueroa Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria,
More informationTriangle deletion. Ernie Croot. February 3, 2010
Triangle deletion Ernie Croot February 3, 2010 1 Introduction The purpose of this note is to give an intuitive outline of the triangle deletion theorem of Ruzsa and Szemerédi, which says that if G = (V,
More informationmost 4 Mirka Miller 1,2, Guillermo PinedaVillavicencio 3, The University of Newcastle Callaghan, NSW 2308, Australia University of West Bohemia
Complete catalogue of graphs of maimum degree 3 and defect at most 4 Mirka Miller 1,2, Guillermo PinedaVillavicencio 3, 1 School of Electrical Engineering and Computer Science The University of Newcastle
More informationFull and Complete Binary Trees
Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full
More informationCMSC 451: Graph Properties, DFS, BFS, etc.
CMSC 451: Graph Properties, DFS, BFS, etc. Slides By: Carl Kingsford Department of Computer Science University of Maryland, College Park Based on Chapter 3 of Algorithm Design by Kleinberg & Tardos. Graphs
More informationMean RamseyTurán numbers
Mean RamseyTurán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρmean coloring of a graph is a coloring of the edges such that the average
More informationAn inequality for the group chromatic number of a graph
An inequality for the group chromatic number of a graph HongJian Lai 1, Xiangwen Li 2 and Gexin Yu 3 1 Department of Mathematics, West Virginia University Morgantown, WV 26505 USA 2 Department of Mathematics
More informationP. Jeyanthi and N. Angel Benseera
Opuscula Math. 34, no. 1 (014), 115 1 http://dx.doi.org/10.7494/opmath.014.34.1.115 Opuscula Mathematica A TOTALLY MAGIC CORDIAL LABELING OF ONEPOINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel
More informationA CHARACTERIZATION OF LOCATINGTOTAL DOMINATION EDGE CRITICAL GRAPHS
Discussiones Mathematicae Graph Theory 31 (2011) 197 202 Note A CHARACTERIZATION OF OCATINGTOTA DOMINATION EDGE CRITICA GRAPHS Mostafa Blidia AMDARO, Department of Mathematics University of Blida, B.P.
More informationDiscrete Applied Mathematics. The firefighter problem with more than one firefighter on trees
Discrete Applied Mathematics 161 (2013) 899 908 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The firefighter problem with
More informationMultilayer Structure of Data Center Based on Steiner Triple System
Journal of Computational Information Systems 9: 11 (2013) 4371 4378 Available at http://www.jofcis.com Multilayer Structure of Data Center Based on Steiner Triple System Jianfei ZHANG 1, Zhiyi FANG 1,
More information(Vertex) Colorings. We can properly color W 6 with. colors and no fewer. Of interest: What is the fewest colors necessary to properly color G?
Vertex Coloring 2.1 33 (Vertex) Colorings Definition: A coloring of a graph G is a labeling of the vertices of G with colors. [Technically, it is a function f : V (G) {1, 2,...,l}.] Definition: A proper
More informationFrans J.C.T. de Ruiter, Norman L. Biggs Applications of integer programming methods to cages
Frans J.C.T. de Ruiter, Norman L. Biggs Applications of integer programming methods to cages Article (Published version) (Refereed) Original citation: de Ruiter, Frans and Biggs, Norman (2015) Applications
More information2partitiontransitive tournaments
partitiontransitive tournaments Barry Guiduli András Gyárfás Stéphan Thomassé Peter Weidl Abstract Given a tournament score sequence s 1 s s n, we prove that there exists a tournament T on vertex set
More information1 Digraphs. Definition 1
1 Digraphs Definition 1 Adigraphordirected graphgisatriplecomprisedofavertex set V(G), edge set E(G), and a function assigning each edge an ordered pair of vertices (tail, head); these vertices together
More informationA CHARACTERIZATION OF TREE TYPE
A CHARACTERIZATION OF TREE TYPE LON H MITCHELL Abstract Let L(G) be the Laplacian matrix of a simple graph G The characteristic valuation associated with the algebraic connectivity a(g) is used in classifying
More informationThe degree, size and chromatic index of a uniform hypergraph
The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a kuniform hypergraph in which no two edges share more than t common vertices, and let D denote the
More informationPlanar embeddability of the vertices of a graph using a fixed point set is NPhard
Planar embeddability of the vertices of a graph using a fixed point set is NPhard Sergio Cabello institute of information and computing sciences, utrecht university technical report UUCS2003031 wwwcsuunl
More informationGraph Theory. Introduction. Distance in Graphs. Trees. Isabela Drămnesc UVT. Computer Science Department, West University of Timişoara, Romania
Graph Theory Introduction. Distance in Graphs. Trees Isabela Drămnesc UVT Computer Science Department, West University of Timişoara, Romania November 2016 Isabela Drămnesc UVT Graph Theory and Combinatorics
More informationCombinatorics: The Fine Art of Counting
Combinatorics: The Fine Art of Counting Week 9 Lecture Notes Graph Theory For completeness I have included the definitions from last week s lecture which we will be using in today s lecture along with
More informationDivisor graphs have arbitrary order and size
Divisor graphs have arbitrary order and size arxiv:math/0606483v1 [math.co] 20 Jun 2006 Le Anh Vinh School of Mathematics University of New South Wales Sydney 2052 Australia Abstract A divisor graph G
More informationOn the Randić index and Diameter of Chemical Graphs
On the Rić index Diameter of Chemical Graphs Dragan Stevanović PMF, University of Niš, Niš, Serbia PINT, University of Primorska, Koper, Slovenia (Received January 4, 008) Abstract Using the AutoGraphiX
More information/ Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry
600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry 2.1 Steiner Tree Definition 2.1.1 In the Steiner Tree problem the input
More informationPlanar Tree Transformation: Results and Counterexample
Planar Tree Transformation: Results and Counterexample Selim G Akl, Kamrul Islam, and Henk Meijer School of Computing, Queen s University Kingston, Ontario, Canada K7L 3N6 Abstract We consider the problem
More informationAn important class of codes are linear codes in the vector space Fq n, where F q is a finite field of order q.
Chapter 3 Linear Codes An important class of codes are linear codes in the vector space Fq n, where F q is a finite field of order q. Definition 3.1 (Linear code). A linear code C is a code in Fq n for
More informationChapter 2 Paths and Searching
Chapter 2 Paths and Searching Section 2.1 Distance Almost every day you face a problem: You must leave your home and go to school. If you are like me, you are usually a little late, so you want to take
More informationThe decay number and the maximum genus of diameter 2 graphs
Discrete Mathematics 226 (2001) 191 197 www.elsevier.com/locate/disc The decay number and the maximum genus of diameter 2 graphs HungLin Fu a; ;1, MingChun Tsai b;1, N.H. Xuong c a Department of Applied
More informationTotal colorings of planar graphs with small maximum degree
Total colorings of planar graphs with small maximum degree Bing Wang 1,, JianLiang Wu, SiFeng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong
More informationCycles and cliqueminors in expanders
Cycles and cliqueminors in expanders Benny Sudakov UCLA and Princeton University Expanders Definition: The vertex boundary of a subset X of a graph G: X = { all vertices in G\X with at least one neighbor
More informationEvery tree is 3equitable
Discrete Mathematics 220 (2000) 283 289 www.elsevier.com/locate/disc Note Every tree is 3equitable David E. Speyer a, Zsuzsanna Szaniszlo b; a Harvard University, USA b Department of Mathematical Sciences,
More informationarxiv:1203.1525v1 [math.co] 7 Mar 2012
Constructing subset partition graphs with strong adjacency and endpoint count properties Nicolai Hähnle haehnle@math.tuberlin.de arxiv:1203.1525v1 [math.co] 7 Mar 2012 March 8, 2012 Abstract Kim defined
More informationA Turán Type Problem Concerning the Powers of the Degrees of a Graph
A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of HaifaORANIM, Tivon 36006, Israel. AMS Subject Classification:
More informationGRAPH THEORY and APPLICATIONS. Trees
GRAPH THEORY and APPLICATIONS Trees Properties Tree: a connected graph with no cycle (acyclic) Forest: a graph with no cycle Paths are trees. Star: A tree consisting of one vertex adjacent to all the others.
More informationGraph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency.
Mária Markošová Graph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency. Isomorphism of graphs. Paths, cycles, trials.
More information