# On a tree graph dened by a set of cycles

Save this PDF as:

Size: px
Start display at page:

## Transcription

2 304 X. Li et al. / Discrete Mathematics 271 (2003) G, and Heinrich and Liu [4] proved that T a (G) is2-connected for any simple graph G. In[3], Harary et al. proved that T l (G) is connected for every 2-connected graph G, and in [1] Broersma and Li characterized those graphs for which T l (G) is connected. For any set of cycles C of a connected graph G we dene T(G; C) as the spanning subgraph of T (G), in which two trees R and S are adjacent if they are adjacent in T(G) and the unique cycle contained in R S lies in C. In this article, we present necessary conditions and sucient conditions for T(G; C) to be connected. Throughout this article we shall denote with the same characters graphs and their sets of edges. The symmetric dierence of two cycles and of a graph G is the subgraph of G induced by the edge set ( )\( ). 2. Necessary conditions For any graph G we denote by (G) the cycle space of G, and for any set C of cycles of G we denote by Span C the subspace of (G) spanned by C. A cycle is cyclically spanned by C if there are cycles 1 ; 2 ;:::; m C such that = 1 2 m and for j =1; 2;:::;m, 1 2 j is a cycle of G. Lemma 2.1. Let C be a set of cycles of a connected graph G and let Q 0 ;Q 1 ;:::;Q n be a path in T (G; C) with length n 1. For i =1; 2;:::;n, denote by i the unique cycle contained in Q i 1 Q i. If is a cycle of G such that Q 0 + e for some e Q n, then is cyclically spanned by { 1 ; 2 ;:::; n }. Proof. If n = 1, then e Q 1 and Q 0 Q 1 which implies = 1. We proceed by induction assuming n 1 and that the result holds for any path in T(G; C) with length less than n. Case 1: e Q t for some t n. Since Q 0 ;Q 1 ;:::;Q t is a path in T(G; C) with length t n, Q 0 + e and e Q t, then by induction is cyclically spanned by { 1 ; 2 ;:::; t }. Case 2: Q t + e for some t 1. Since Q t ;Q t+1 ;:::;Q n is a path in T (G; C) with length n t n, Q t + e and e Q n, then by induction is cyclically spanned by { t+1 ; t+2 ;:::; n }. Case 3: * Q t + e for t =1; 2;:::;n and e Q t for t =0; 1;:::;n 1. Let a and b be the edges of G such that Q 1 is obtained from Q 0 by deleting a and adding b. Clearly a; b 1, and since is contained in Q 0 + e but not in Q 1 + e =((Q 0 a)+b)+e, then a also lies in. Since Q 0 + e and 1 Q 0 Q 1, then 1 (Q 0 Q 1 )+e =(Q 1 + a)+e. Since a 1, then 1 Q 1 + e. In this case Q 1 ;Q 2 ;:::;Q n is a path in T(G; C) with length n 1, 1 Q 1 +e and e Q n. By induction 1 is cyclically spanned by { 2 ; 3 ;:::; n }. Since =( 1 ) 1, then is cyclically spanned by { 1 ; 2 ;:::; n }. Theorem 2.2. Let C be a set of cycles of a connected graph G. If T(G; C) is connected, then C spans the cycle space of G.

3 X. Li et al. / Discrete Mathematics 271 (2003) Proof. Let be any cycle of G and e be an edge of. Let R and S be spanning trees of G such that R + e and e S. Since T(G; C) is connected, there is a path R = Q 0 ;Q 1 ;:::;Q n = S connecting R and S in T(G; C). By Lemma 2.1, Span{ 1 ; 2 ;:::; n }, where for i =1; 2;:::;n, i C is the unique cycle of G contained in Q i 1 Q i. Consider the complete graph G with vertex set {u 1 ;u 2 ;u 3 ;u 4 }. Let C consist of the cycles u 1 u 2 u 3 u 4, u 1 u 2 u 4 and u 1 u 4 u 3. The set C is a basis of the cycle space of G, nevertheless the path u 1 u 3 u 2 u 4 is an isolated vertex in T(G; C). This shows that the condition on Theorem 2.2 is not a sucient condition for T(G; C) to be connected. 3. Sucient conditions A unicycle of a connected graph G is a connected spanning subgraph of G that contains exactly one cycle. Let C be a set of cycles of a connected graph G. A cycle of G satises property (with respect to C) if for any unicycle U of G containing there are two cycles ; C contained in U + e for some edge e of G such that =. Lemma 3.1. Let C be a set of cycles of a connected graph G and be a cycle of G satisfying property. The graph T (G; C) is connected if and only if T(G; C {}) is connected. Proof. Clearly T (G; C {}) is connected whenever T(G; C) is connected. Let R and S be spanning trees of G, adjacent in T(G; C {}) and! C {} be the unique cycle contained in R S. If! C, then R and S are also adjacent in T(G; C). Suppose now! =. Since satises property and R S is a unicycle of G that contains, there are two cycles ; C contained in (R S)+e for some edge e of G such that =. Let a and b be the edges of such that S =(R a)+b. Four cases are considered. Case 1: a \ and b \. Let Q be the spanning tree of G obtained from R by deleting the edge a and adding the edge e. Since S =(Q e)+b, both pairs of trees R and Q and Q and S are adjacent in T(G). Since (R S)+e =(R + b)+e and b, then R + e = R Q; therefore R and Q are adjacent in T (G; {}). Since (R S)+e =(S + a)+e and a, then S + e = S Q; therefore S and Q are adjacent in T (G; {}). In this case R and S are connected in T(G; {; }) T(G; C) by a path of length two. Case 2: b \ and a \. Interchange and in Case 1. Case 3: a; b \.

4 306 X. Li et al. / Discrete Mathematics 271 (2003) Let c be an edge in \ and let Q 1 =(R c) +e and Q 2 =(S c) +e. Since Q 2 =(Q 1 a)+b, all three pairs R and Q 1, Q 1 and Q 2 and Q 2 and S are adjacent in T(G). Since (R S)+e =(R + b)+e and b, then R + e = R Q 1 ; therefore R and Q 1 are adjacent in T (G; {}). Since (R S)+e =(Q 1 Q 2 )+c and c, then Q 1 Q 2 ; therefore Q 1 and Q 2 are adjacent in T (G; {}). Since (R S)+e =(S + a)+e and a, then S + e = Q 2 S; therefore Q 2 and S are adjacent in T (G; {}). In this case, R and S are connected in T(G; {; }) T(G; C) by a path of length three. Case 4: a; b \. Interchange and in Case 3. Let G be a connected graph. For any set C of cycles of G, we dene the closure cl G (C) ofc in G as the set of cycles obtained from C by recursively adding new cycles of G that satisfy property until no such cycle remains. Theorem 3.2. For any connected graph G and any set C of cycles of G, the closure of C in G is well dened. Proof. Suppose the result is false and let C and C be two dierent sets of cycles of G obtained from C by recursively adding new cycles of G that satisfy property until no such cycle remains. Let 1 ; 2 ;:::; n and 1 ; 2 ;:::; m denote the sequences of cycles added to C while obtaining C and C, respectively. Without loss of generality we assume C = C and let k be the rst cycle in the sequence 1 ; 2 ;:::; n which is not in C. Let D = C { 1 ; 2 ;:::; k 1 }; since k satises property with respect to D and D C, then k satises property with respect to C which is not possible since C is -closed and k C. Theorem 3.3. Let C be a set of cycles of a connected graph G. The graph T(G; C) is connected if and only if T (G; cl G (C)) is connected. Proof. Let 1 ; 2 ;:::; n be the sequence of cycles added to C while obtaining cl G (C). Set C 0 = C and for i =0; 1;:::;n 1, let C i+1 = C i { i+1 }. Clearly C n = cl G (C) and by Lemma 3.1, T (G; C i ) is connected if and only if T(G; C i+1 ) is connected. A set of cycles C of a connected graph G is -dense in G if cl G (C) is the set of all cycles of G. Corollary 3.4. If C is a -dense set of cycles in a connected graph G, then T(G; C) is connected. Proof. If C is -dense, then T (G; cl G (C)) = T(G) which is always connected. By Theorem 3.3, T (G; C) is connected.

5 X. Li et al. / Discrete Mathematics 271 (2003) We know no example of a connected graph G and a set of cycles C of G such that T(G; C) is connected but C is not -dense in G. 4. -dense sets of cycles In this section, we present two examples of sets of cycles which are -dense. Theorem 4.1. For any 2-connected plane graph G, the set C of internal faces of G is -dense in G. Proof. Let be a cycle of G and k be the number of edges of G contained in the interior of. Ifk = 0, then C cl G (C). We proceed by induction assuming k 1 and that if is a cycle of G whose interior contains fewer than k edges of G, then cl G (C). Let U be a unicycle of G containing. For each vertex w of G let U w be the minimal path contained in U that connects w to and denote by w() the unique vertex of U w that lies in. Since k 1 and G is 2-connected, there is an edge e = uv of G, contained in the interior of, such that u() v(). Let L and R the two paths contained in, joining u() and v() and let =(U u U v L)+e and =(U u U v R)+e. Since u() v(), then U u and U v are disjoint paths and therefore and are cycles of G contained in U + e. Moreover, since G is a plane graph and e is contained in the interior of, all edges of U u U v are also contained in the interior of ; this implies that the interiors of and are contained in the interior of and therefore contain fewer than k edges of G. By induction and must be in cl G (C). Since = L R =((U u U v L)+e)((U u U v R)+e)=, then satises property with respect to cl G (C) and therefore cl G (C). Corollary 4.2. If C is the set of internal faces of a 2-connected plane graph G, then T(G; C) is connected. Theorem 4.3. Let e = uv be an edge of a 2-connected graph G. If C e is the set of cycles of G that contain the edge e, then C e is -dense in G. Proof. For every path L in G we denote by l(l) the length of L. Assume the result is false and for each cycle of G, not in cl G (C e ), let L and R be disjoint paths of G connecting u and v to, respectively, and such that l(l ) l(l), or l(l )=l(l) and l(r ) 6 l(r) for any pair L and R of disjoint paths of G that connect u and v to, respectively. Choose (G)\cl G (C e ) such that l(l ) l(l ), or l(l ) = l(l ) and l(r ) 6 l(r ) for any cycle (G)\cl G (C e ). Let u = u 0 ;u 1 ;:::;u n be the path L and v = v 0 ;v 1 ;:::;v m be the path R.

6 308 X. Li et al. / Discrete Mathematics 271 (2003) Let U be a unicycle of G containing. As in Theorem 4.1, for each vertex w of G let U w be the minimal path contained in U that connects w to and denote by w() the unique vertex of U w that lies in. Case 1: u() v(). Denote by A and B the two paths contained in joining u() and v() and let =(U u U v A)+e and =(U u U v B)+e. Since u() v(), then U u and U v are disjoint paths and therefore and are cycles of G contained in U + e. Since the edge e belongs to both cycles and, then ; C e and since =((U u U v A)+ e)((u u U v B) +e) =A B =, then satises property with respect to C which is a contradiction. Case 2: u() =v(). Since L and R are disjoint paths, either u n u() orv m v(). Subcase 2.1: u n u(). Since u n ()=u n u()=u 0 (), there is an edge f=u i u i+1 in L such that u i ()=u() and u i+1 () u(). In this case let =(U ui U ui+1 Q)+f and =(U ui U ui+1 R)+f, where Q and R are the two paths contained in, joining u i () and u i+1 (). Since U ui and U ui+1 are disjoint paths, and are cycles of G. Since u = u 0 ;u 1 ;:::;u i is a path in G, with length i n= l(l ), joining u to both cycles and, then l(l ) l(l ) and l(l ) l(l ). By the choice of, both cycles and are in cl G (C e ). Since and are contained in U + f and =, then satises property with respect to cl G (C e ) which is a contradiction. Subcase 2.2: u n = u() and v m v(). Since v m ()=v m v()=v 0 (), there is an edge g=v i v i+1 in R such that v i ()=v() and v i+1 () v(). In this case, let =(U vi U vi+1 Q)+g and =(U vi U vi+1 R)+g, where Q and R are the two paths contained in, joining v i () and v i+1 (). Since U vi and U vi+1 are disjoint paths, and are cycles of G. Since u n = u() =v() =v i (), then u n lies in U vi. This implies that u = u 0 ;u 1 ;:::;u n is a path of length n = l(l ) that joins u to and to. Therefore, l(l ) 6 l(l ) and l(l ) 6 l(l ). Since v=v 0 ;v 1 ;:::;v i is a path in G, with length i m, joining v to both cycles and, then l(r ) 6 i m=l(r ) and l(r ) 6 i m=l(r ). By the choice of, both cycles and must be in cl G (C e ). Since and are contained in U + g and =, then satises property with respect to cl G (C e ) which, again, is a contradiction. Corollary 4.4. Let e be an edge of a 2-connected graph G. If C e is the set of cycles of G that contain the edge e, then T (G; C) is connected. 5. The basis graph of a binary matroid A binary matroid is a matroid M such that for any two circuits and, the symmetric dierence contains a circuit. A matroid is loopless if it has no circuit consisting of a single element. The basis graph of a binary matroid M is the graph B(M) whose vertices are the basis of M, in which two basis R and S are adjacent if S can be obtained from R by deleting an element r of R and adding an another element s of S.

7 X. Li et al. / Discrete Mathematics 271 (2003) For any set C of circuits of a binary matroid M, we dene a graph B(M; C) in which two basis R and S are adjacent if they are adjacent in B(M) and the unique circuit of M contained in R S lies in C. A unicircuit of a loopless binary matroid M is a set obtained from a basis of M by adding a new element. Let C be a set of circuits of a loopless binary matroid M. A circuit of M satises property (with respect to C) if for any unicircuit U of M containing, there are two circuits ; C contained in U + e for some element e of M such that =. As for a set of cycles in a graph, we can dene the closure cl M (C) of a set of circuits C in a loopless binary matroid M as the set of circuits of M obtained from C by adding new circuits of M that satisfy property until no such circuit remains. A set of circuits C is -dense in M if cl M (C) contains every circuit of M. The following results can be proved in an analogous way as the corresponding results for graphs. Theorem 5.1. Let C be a set of circuits of a loopless binary matroid M. If B(M; C) is connected, then C spans the circuit space of M. Theorem 5.2. IfCisa -dense set of circuits in a loopless binary matroid M, then B(M; C) is connected. Let F be graph and X be a set of vertices of F; we denote by F[X ] the subgraph of F induced by X. For any disjoint sets X and Y of vertices of F let [X; Y ] denote the set of edges of F joining a vertex in X with a vertex in Y.Abond of a connected graph F is a set [X; X ] such that both graphs F[X ] and F[ X ] are connected, where X = V (F)\X. Another example of a -dense set of circuits worth to mention is the following: Let G be a 2-connected graph and M(G) be the cographic matroid of G, where the circuits are the bonds of G and the basis are the complements of the spanning trees of G. Let v 1 ;v 2 ;:::;v n denote the vertices of G and for i =1; 2;:::;n let i be the set of edges incident with v i. Since G is 2-connected, i is a bond of G for i =1; 2;:::;n; let C = { 1 ; 2 ;:::; n }. The graph B(M(G);C) is isomorphic to the leaf exchange graph T l (G) and therefore it is connected. We claim that C is -dense in M(G). Moreover, the set C n = { 1 ; 2 ;:::; n 1 } is -dense in M(G). Proof of claim. Let =[X; X ] be any bond of G and assume without loss of generality that v n X.If X = 1, then C n cl M(G) (C n ). We proceed by induction assuming X 1 and that if =[Y; Y ] is any bond of G with v n Y and Y X, then cl M(G) (C n ). Let U be a unicircuit of M(G) containing and let B be a basis of M(G) such that U = B {x} for some edge x of G. Then U = T + x for some edge x of T, where T is the spanning tree of G such that B = T. Since T + x, then x is the only edge of T contained in and therefore T [X ] and T[ X ] are spanning trees of G[X ] and G[ X ], respectively.

8 310 X. Li et al. / Discrete Mathematics 271 (2003) For each edge c let u c denote the end of c in X. Since G is 2-connected and X 1, there are two edges a; b such that u a u b. Let P be the unique path contained in T [X ] that joins u a and u b and let e be any edge of P. Let X a and X b denote the sets of vertices in X which are connected in T[X ] e to u a and to u b, respectively and let =[X a ; X X b ] and =[X b ; X X a ]. Since X X b =X a, X X a = X b and T [X a ], T [X b ], (T [ X ] T [X a ]) + a and (T[ X ] T[X b ]) + b are spanning trees of G[X a ], G[X b ], G[ X X a ] and G[ X X b ], respectively, then and are bonds of G. Since =[X a ; X X b ]=[X a ; X ] [X a ;X b ], [X a ; X ] T + x and [X a ;X b ] T[X ]+e T + e, then ( T + x)+e = U + e. Analogously U + e. By induction ; cl M(G) (C n ), since X a X and X b X. Notice that =[X a ; X X b ][X b ; X X a ] =([X a ; X ] [X a ;X b ])([X b ; X ] [X b ;X a ]) =[X a ; X ] [X b ; X ] =[X a X b ; X ] =[X; X ] = ; hence satises property with respect to cl M(G) (C n ) and therefore cl M(G) (C n ). References [1] H.J. Broersma, X. Li, The connectivity of the leaf-exchange spanning tree graph of a graph, Ars Combin. 43 (1996) [2] R.L. Cummins, Hamilton circuits in tree graphs, IEEE Trans. Circuit Theory CT-13 (1966) [3] F. Harary, R.J. Mokken, M.J. Plantholt, Interpolation theorem for diameters of spanning trees, IEEE Trans. Circuits Systems 30 (7) (1983) [4] K. Heinrich, G. Liu, A lower bound on the number of spanning trees with k endvertices, J. Graph Theory 12 (1) (1988) [5] C.A. Holzmann, F. Harary, On the tree graph of a matroid, SIAM J. Appl. Math. 22 (1972) [6] F.J. Zhang, Z. Chen, Connectivity of (adjacency) tree graphs, J. Xinjiang Univ. Natur. Sci. 3 (4) (1986) 1 5.

### ON 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,

### A counterexample to a result on the tree graph of a graph

AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 63(3) (2015), Pages 368 373 A counterexample to a result on the tree graph of a graph Ana Paulina Figueroa Departamento de Matemáticas Instituto Tecnológico

### A note on properties for a complementary graph and its tree graph

A note on properties for a complementary graph and its tree graph Abulimiti Yiming Department of Mathematics Institute of Mathematics & Informations Xinjiang Normal University China Masami Yasuda Department

### A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

### The 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 Hung-Lin Fu a; ;1, Ming-Chun Tsai b;1, N.H. Xuong c a Department of Applied

### On the k-path cover problem for cacti

On the k-path 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

### Unicyclic 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

### Math 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

### Characterizations 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

### Solutions 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.

### 1 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

### Tenacity and rupture degree of permutation graphs of complete bipartite graphs

Tenacity and rupture degree of permutation graphs of complete bipartite graphs Fengwei Li, Qingfang Ye and Xueliang Li Department of mathematics, Shaoxing University, Shaoxing Zhejiang 312000, P.R. China

### 4 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

### An 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 Hong-Jian Lai a, Xiangwen Li b,,1, Gexin Yu c a Department of Mathematics,

### Every tree is 3-equitable

Discrete Mathematics 220 (2000) 283 289 www.elsevier.com/locate/disc Note Every tree is 3-equitable David E. Speyer a, Zsuzsanna Szaniszlo b; a Harvard University, USA b Department of Mathematical Sciences,

### Connectivity 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

### Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

### Basic Notions on Graphs. Planar Graphs and Vertex Colourings. Joe Ryan. Presented by

Basic Notions on Graphs Planar Graphs and Vertex Colourings Presented by Joe Ryan School of Electrical Engineering and Computer Science University of Newcastle, Australia Planar graphs Graphs may be drawn

### An inequality for the group chromatic number of a graph

An inequality for the group chromatic number of a graph Hong-Jian Lai 1, Xiangwen Li 2 and Gexin Yu 3 1 Department of Mathematics, West Virginia University Morgantown, WV 26505 USA 2 Department of Mathematics

### Graph 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

### THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER

THE 0/1-BORSUK 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

### Chapter 4: Trees. 2. Theorem: Let T be a graph with n vertices. Then the following statements are equivalent:

9 Properties of Trees. Definitions: Chapter 4: Trees forest - a graph that contains no cycles tree - a connected forest. Theorem: Let T be a graph with n vertices. Then the following statements are equivalent:

### On Maximal Energy of Trees with Fixed Weight Sequence

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 75 (2016) 267-278 ISSN 0340-6253 On Maximal Energy of Trees with Fixed Weight Sequence Shicai Gong 1,,,

### SHARP 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.,

### Diameter of paired domination edge-critical graphs

AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 40 (2008), Pages 279 291 Diameter of paired domination edge-critical graphs M. Edwards R.G. Gibson Department of Mathematics and Statistics University of Victoria

### Divisor 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

### Chapter 4. Trees. 4.1 Basics

Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.

### Degree-associated reconstruction parameters of complete multipartite graphs and their complements

Degree-associated reconstruction parameters of complete multipartite graphs and their complements Meijie Ma, Huangping Shi, Hannah Spinoza, Douglas B. West January 23, 2014 Abstract Avertex-deleted subgraphofagraphgisacard.

### Odd 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 long-standing

### GRAPH 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.

### COLORED GRAPHS AND THEIR PROPERTIES

COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring

### most 4 Mirka Miller 1,2, Guillermo Pineda-Villavicencio 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 Pineda-Villavicencio 3, 1 School of Electrical Engineering and Computer Science The University of Newcastle

### Planar 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

### Homework 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

### A 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

### Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs

MCS-236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set

### Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

### 10. 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

### Graph 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.

### GRAPH THEORY LECTURE 4: TREES

GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection

### Edge looseness of plane graphs

Also available at http://amc-journal.eu ISSN 1855-966 (printed edn.), ISSN 1855-974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (015) 89 96 Edge looseness of plane graphs Július Czap Department of

### 3. 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

### P. 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 ONE-POINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel

### Special Classes of Divisor Cordial Graphs

International Mathematical Forum, Vol. 7,, no. 35, 737-749 Special Classes of Divisor Cordial Graphs R. Varatharajan Department of Mathematics, Sri S.R.N.M. College Sattur - 66 3, Tamil Nadu, India varatharajansrnm@gmail.com

### Connections between decomposition trees of 3-connected plane triangulations and hamiltonian properties

Introduction Hamiltonicity Other hamiltonian properties Connections between decomposition trees of 3-connected plane triangulations and hamiltonian properties Gunnar Brinkmann Jasper Souffriau Nico Van

### Planarity Planarity

Planarity 8.1 71 Planarity Up until now, graphs have been completely abstract. In Topological Graph Theory, it matters how the graphs are drawn. Do the edges cross? Are there knots in the graph structure?

### On end degrees and infinite cycles in locally finite graphs

On end degrees and infinite cycles in locally finite graphs Henning Bruhn Maya Stein Abstract We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel

### Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices

MATHEMATICAL COMMUNICATIONS 47 Math. Commun., Vol. 15, No. 2, pp. 47-58 (2010) Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices Hongzhuan Wang 1, Hongbo Hua 1, and Dongdong Wang

### Chapter 6 Planarity. Section 6.1 Euler s Formula

Chapter 6 Planarity Section 6.1 Euler s Formula In Chapter 1 we introduced the puzzle of the three houses and the three utilities. The problem was to determine if we could connect each of the three utilities

### Graphs 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

### 1 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

### Lattice Point Geometry: Pick s Theorem and Minkowski s Theorem. Senior Exercise in Mathematics. Jennifer Garbett Kenyon College

Lattice Point Geometry: Pick s Theorem and Minkowski s Theorem Senior Exercise in Mathematics Jennifer Garbett Kenyon College November 18, 010 Contents 1 Introduction 1 Primitive Lattice Triangles 5.1

### Total colorings of planar graphs with small maximum degree

Total colorings of planar graphs with small maximum degree Bing Wang 1,, Jian-Liang Wu, Si-Feng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong

### On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

### Extremal Wiener Index of Trees with All Degrees Odd

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 287-292 ISSN 0340-6253 Extremal Wiener Index of Trees with All Degrees Odd Hong Lin School of

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

Available online at www.sciencedirect.com ScienceDirect Procedia Computer Science 74 (05 ) 47 5 International Conference on Graph Theory and Information Security Fractional Metric Dimension of Tree and

### Two General Methods to Reduce Delay and Change of Enumeration Algorithms

ISSN 1346-5597 NII Technical Report Two General Methods to Reduce Delay and Change of Enumeration Algorithms Takeaki Uno NII-2003-004E Apr.2003 Two General Methods to Reduce Delay and Change of Enumeration

### Theorem A graph T is a tree if, and only if, every two distinct vertices of T are joined by a unique path.

Chapter 3 Trees Section 3. Fundamental Properties of Trees Suppose your city is planning to construct a rapid rail system. They want to construct the most economical system possible that will meet the

### 1 Basic Definitions and Concepts in Graph Theory

CME 305: Discrete Mathematics and Algorithms 1 Basic Definitions and Concepts in Graph Theory A graph G(V, E) is a set V of vertices and a set E of edges. In an undirected graph, an edge is an unordered

### R u t c o r Research R e p o r t. Boolean functions with a simple certificate for CNF complexity. Ondřej Čepeka Petr Kučera b Petr Savický c

R u t c o r Research R e p o r t Boolean functions with a simple certificate for CNF complexity Ondřej Čepeka Petr Kučera b Petr Savický c RRR 2-2010, January 24, 2010 RUTCOR Rutgers Center for Operations

### 14. Trees and Their Properties

14. Trees and Their Properties This is the first of a few articles in which we shall study a special and important family of graphs, called trees, discuss their properties and introduce some of their applications.

### Tree Cover of Graphs

Applied Mathematical Sciences, Vol. 8, 2014, no. 150, 7469-7473 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.49728 Tree Cover of Graphs Rosalio G. Artes, Jr. and Rene D. Dignos Department

### Tools for parsimonious edge-colouring of graphs with maximum degree three. J.L. Fouquet and J.M. Vanherpe. Rapport n o RR-2010-10

Tools for parsimonious edge-colouring of graphs with maximum degree three J.L. Fouquet and J.M. Vanherpe LIFO, Université d Orléans Rapport n o RR-2010-10 Tools for parsimonious edge-colouring of graphs

### SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov

Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices

### Cycles 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 CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

### 1. Relevant standard graph theory

Color identical pairs in 4-chromatic graphs Asbjørn Brændeland I argue that, given a 4-chromatic graph G and a pair of vertices {u, v} in G, if the color of u equals the color of v in every 4-coloring

### UPPER 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

### CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992

CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992 Original Lecture #6: 28 January 1991 Topics: Triangulating Simple Polygons

### Single machine parallel batch scheduling with unbounded capacity

Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University

### Cycle transversals in bounded degree graphs

Electronic Notes in Discrete Mathematics 35 (2009) 189 195 www.elsevier.com/locate/endm Cycle transversals in bounded degree graphs M. Groshaus a,2,3 P. Hell b,3 S. Klein c,1,3 L. T. Nogueira d,1,3 F.

### arxiv: 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

### Trees and Fundamental Circuits

Trees and Fundamental Circuits Tree A connected graph without any circuits. o must have at least one vertex. o definition implies that it must be a simple graph. o only finite trees are being considered

### Non-Separable Detachments of Graphs

Egerváry Research Group on Combinatorial Optimization Technical reports TR-2001-12. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.

### DO NOT RE-DISTRIBUTE 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

### Combinatorics: 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

### Coloring Eulerian triangulations of the projective plane

Coloring Eulerian triangulations of the projective plane Bojan Mohar 1 Department of Mathematics, University of Ljubljana, 1111 Ljubljana, Slovenia bojan.mohar@uni-lj.si Abstract A simple characterization

### A 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

### Homework Exam 1, Geometric Algorithms, 2016

Homework Exam 1, Geometric Algorithms, 2016 1. (3 points) Let P be a convex polyhedron in 3-dimensional space. The boundary of P is represented as a DCEL, storing the incidence relationships between the

### (a) (b) (c) Figure 1 : Graphs, multigraphs and digraphs. If the vertices of the leftmost figure are labelled {1, 2, 3, 4} in clockwise order from

4 Graph Theory Throughout these notes, a graph G is a pair (V, E) where V is a set and E is a set of unordered pairs of elements of V. The elements of V are called vertices and the elements of E are called

### 7. Colourings. 7.1 Vertex colouring

7. Colourings Colouring is one of the important branches of graph theory and has attracted the attention of almost all graph theorists, mainly because of the four colour theorem, the details of which can

### Discrete Mathematics Problems

Discrete Mathematics Problems William F. Klostermeyer School of Computing University of North Florida Jacksonville, FL 32224 E-mail: wkloster@unf.edu Contents 0 Preface 3 1 Logic 5 1.1 Basics...............................

### 136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics

### Hamiltonian Claw-free Graphs and Line Graphs

Hamiltonian Claw-free Graphs and Line Graphs Hong-Jian Lai Department of Mathematics, West Virginia University Joint work with Ye Chen, Keke Wang and Meng Zhang p. 1/22 Contents The Problem p. 2/22 Contents

### RAMSEY FOR COMPLETE GRAPHS WITH DROPPED CLIQUES

RAMSEY FOR COMPLETE GRAPHS WITH DROPPED CLIQUES JONATHAN CHAPPELON, LUIS PEDRO MONTEJANO, AND JORGE LUIS RAMÍREZ ALFONSÍN Abstract Let K [k,t] be the complete graph on k vertices from which a set of edges,

### All 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

### HOMEWORK #3 SOLUTIONS - MATH 3260

HOMEWORK #3 SOLUTIONS - MATH 3260 ASSIGNED: FEBRUARY 26, 2003 DUE: MARCH 12, 2003 AT 2:30PM (1) Show either that each of the following graphs are planar by drawing them in a way that the vertices do not

### A graphical characterization of the group Co 2

A graphical characterization of the group Co 2 Hans Cuypers Department of Mathematics Eindhoven University of Technology Postbox 513, 5600 MB Eindhoven The Netherlands 18 September 1992 Abstract In this

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

### Graph theoretic techniques in the analysis of uniquely localizable sensor networks

Graph theoretic techniques in the analysis of uniquely localizable sensor networks Bill Jackson 1 and Tibor Jordán 2 ABSTRACT In the network localization problem the goal is to determine the location of

### Every 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

### 6.042/18.062J Mathematics for Computer Science October 3, 2006 Tom Leighton and Ronitt Rubinfeld. Graph Theory III

6.04/8.06J Mathematics for Computer Science October 3, 006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Graph Theory III Draft: please check back in a couple of days for a modified version of these

### A CHARACTERIZATION OF LOCATING-TOTAL DOMINATION EDGE CRITICAL GRAPHS

Discussiones Mathematicae Graph Theory 31 (2011) 197 202 Note A CHARACTERIZATION OF OCATING-TOTA DOMINATION EDGE CRITICA GRAPHS Mostafa Blidia AMDA-RO, Department of Mathematics University of Blida, B.P.

### Chapter I Logic and Proofs

MATH 1130 1 Discrete Structures Chapter I Logic and Proofs Propositions A proposition is a statement that is either true (T) or false (F), but or both. s Propositions: 1. I am a man.. I am taller than

### Generalizing 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

### Topology-based network security

Topology-based network security Tiit Pikma Supervised by Vitaly Skachek Research Seminar in Cryptography University of Tartu, Spring 2013 1 Introduction In both wired and wireless networks, there is the

### Extremal 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

### On the Ramsey Number R(4, 6)

On the Ramsey Number R(4, 6) Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809 ge@cs.indstate.edu Submitted: Mar 6, 2012; Accepted: Mar 23, 2012;

### Graph 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

### SCORE SETS IN ORIENTED GRAPHS

Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in

### Computer Science Department. Technion - IIT, Haifa, Israel. Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

- 1 - THREE TREE-PATHS Avram Zehavi Alon Itai Computer Science Department Technion - IIT, Haifa, Israel Abstract Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there