On onefactorizations of replacement products


 Paul Norton
 2 years ago
 Views:
Transcription
1 Filomat 27:1 (2013), DOI /FIL A Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: On onefactorizations of replacement products Alireza Abdollahi a, Amir Loghman b a Department of Mathematics, University of Isfahan, Isfahan , Iran and School of Mathematics, Institute for Research in Fundamental Sciences, P.O. box , Tehran, Iran. b Department of Mathematics, University of Isfahan, Isfahan , Iran. Abstract. Let be an (n, m)graph (n vertices and mregular) and H be an (m, d)graph. Randomly number the edges around each vertex of by {1,..., m} and fix it. Then the replacement product RH of graphs and H (with respect to the numbering) has vertex set V(RH) = V() V(H) and there is an edge between (v, k) and (w, l) if v = w and kl E(H) or vw E() and kth edge incident on vertex v in is connected to the vertex w and this edge is the lth edge incident on w in, where the numberings k and l refers to the random numberings of edges adjacent to any vertex of. If the set of edges of a graph can be partitioned to a set of complete matchings, then the graph is called 1factorizable and any such partition is called a 1factorization. In this paper, 1factorizability of the replacement product RH of graphs and H is studied. As an application we show that fullerene C 60 and C 4 C 8 nanotorus are 1factorizable. 1. Introduction raphs considered in this paper are finite, simple and undirected. Let be a simple graph. The vertex set, edge set, maximum degree and minimum degree of will be denoted by V(), E(), (), δ() respectively. The order of is V(), i.e. the number of vertices of. A graph is called regular if δ() = () and the latter integer is called the degree of and denoted by d(). A dregular graph on n vertices is called an (n, d)graph. A graph is called cubic if it is 3regular. We denote by K n and C n the complete graph of order n and the cycle of order n, respectively. We denote by N (x) the neighborhoods of x V(), that is, the set of vertices adjacent to x. A spanning subgraph of a graph is a subgraph H of such that V(H) = V(). An rregular spanning subgraph of is called an rfactor. A 1factorization of is a set of edgedisjoint 1factors of whose union is E(). The graph is said to be 1factorizable if it has a 1factorization. A necessary condition for a graph to be 1factorizable is that is a regular graph of even order. The concept of 1factorization can be expressed by the concept of edge coloring. An edge coloring of a graph is a map θ : E() C, where C is a set, called the color set, and θ(e) θ( f ) for any pair e and f of adjacent edges of. If α is a color, the set θ 1 ({α}), i.e. the set of edges of colored α, is called the αcolor class. If C = m we say that θ is an medge coloring. The least integer m for which an medge coloring of exists is called the edge chromatic index of and denoted by χ (). It is easily seen that χ () () for any graph. Vizing [9] proved that 2010 Mathematics Subject Classification. Primary 05C15; Secondary 05C70 Keywords. Replacement product; Onefactorization; Fullerene; Nanotorus. Received: 12 April 2011; Accepted: 23 August 2011 Communicated by Dragan Stevanović This research was in part supported by a grant from IPM (No ). This research was also in part supported by the Center of Excellence for Mathematics, University of Isfahan. addresses: (Alireza Abdollahi), (Amir Loghman)
2 A. Abdollahi, A. Loghman / Filomat 27:1 (2013), Theorem 1.1 (Vizing s Theorem). χ () () + 1 for any graph. If in a regular graph, we have χ () = () then edges of each color class consists a 1factor and is 1factorizable. We call a graph Hamiltonian if it contains a spanning cycle. Such a cycle is called a Hamiltonian cycle. Any Hamiltonian graph of even order has a 1factor, for if s a Hamiltonian cycle of and we select alternate edges of C, we eventually end up with a 1factor of. A 2factor all of whose components are even is called an even 2factor. We shall also need the following well known sufficient condition for the existence of a Hamilton cycle in a graph due to Dirac [4]. Theorem 1.2 (Dirac s Theorem). Let be a graph of order at least three such that δ() V() 2. Then is Hamiltonian. Let A and B be finite groups. Assume that B acts on A, namely we are fixing a homomorphism ϕ from B to the automorphism group of A and for elements a A, b B we denote by a b the element a bϕ the action of b on a. We also use a B to denote the orbit of a under this action. The Cayley graph C(H, S) of a group H and a generating set S = S 1 := {s 1 s S}, is an undirected graph whose vertices are the elements of H, and where {, h} is an edge if 1 h S. The Cayley graph C(H, S) is S regular Factorizations of replacement products In this section we describe the replacement product and investigate 1factorizability of replacement product. Let be any (n, k)graph and let [k] = {1,..., k}. By a random numbering of we mean a random numbering of the edges around each vertex of by the numbers in {1,..., k}. More precisely, a random numbering of is a set φ consisting of bijection maps φ x : N (x) [k] for any x V(). Thus the graph has (k!) n random numberings. Example 2.1. Suppose = C(A, S) is a Cayley graph. Then the edges around each vertex of are naturally labeled by the elements of S: if {x, y} E() then φ x (y) = f (x 1 y), where f is a bijection map from S to [ S ]. Definition 2.2. Let be an (n, k)graph and let H be a (k, k )graph with V(H) = [k] = {1,..., k} and fix a random numbering φ of. The replacement product R φ H is the graph whose vertex set is V() V(H) and there is an edge between vertices (v, k) and (w, l) whenever v = w and kl E(H) or vw E(), φ v (w) = k and φw (v) = l. Note that the definition of R φ H clearly depends on φ. Thus for given any two regular graphs and H as above, there are ( V(H)!) V() replacement products which are not necessarily isomorphic. It follows from the definition that R φ H is a regular graph and in fact it is a (nk, k + 1)graph (see [1, 7, 8]). Example 2.3. Let = K 5 and H = C 4 and let φ, φ be as shown in Figure 1. Then the replacement product R φ H and R φ H are not isomorphic, (see Figure 2) since the determinants of the adjacency matrices of R φ H and R φ H are 9 and 12, respectively. Lemma 2.4. Let be an (n, k)graph. Then is 1factorizable if and only if there exists a random numbering φ such that for any edge xx of we have φ x (x ) = φ x (x). Proof. : Let have a 1factorization. Then, since is kregular, there exists an edge coloring θ : E() [k] for. Now define the map φ x from N (x) to [k] for any x V() as φ x (y) = θ(xy) for all y N (x). Clearly φ x is a bijection and we have φ x (x ) = θ(xx ) = θ(x x) = φ x (x)
3 A. Abdollahi, A. Loghman / Filomat 27:1 (2013), Figure 1: = K 5 and H = C 4 Figure 2: (A) is R φ H and (B) is R φ H for any xx E(). Thus φ = {φ x x V()} is the desired random numbering of. : Suppose that has a random numbering φ with the mentioned property in the lemma. Let β be the map from E() to [k] defined by β(xy) = φ x (y) for any xy E(). Since φx (y) = φy (x) (β(xy) = β(yx)), then β is welldefined. Now we show that β is an edge coloring for. To do this, suppose e, f be two adjacent edges of E(). Then there exist x, y, y V() with e = xy, f = xy and y y. Hence φ x (y) φx (y ), as φ x is onetoone. This implies that β(e) β( f ), as required. If is a 1factorizable graph, a random numbering φ is called a 1factorizable numbering whenever for any edge xx of we have φ x (x ) = φ x (x). Thus it follows from Lemma 2.4 that every 1factorizable graph has a 1factorizable numbering. We now consider the case when the two components of the product graph are Cayley graphs = C(A, S A ) and H = C(B, S B ). Furthermore, suppose that B acts on A in such a way that S A = α B for some α S A. So the edges around each vertex of are naturally labeled by the elements of B. This enables us to define the replacement product of and H. The following result is wellknown and we give a proof by using Lemma 2.4.
4 A. Abdollahi, A. Loghman / Filomat 27:1 (2013), Lemma 2.5. Let be a cubic graph. Then the following properties are equivalent: (i) is 1factorizable. (ii) has an even 2factor. (iii) R φ C 3 is 1factorizable for any 1factorizable numbering φ. Proof. (i) (ii) Suppose that is 1factorizable. Then, since is 3regular, there exists an edge coloring θ : E() [3]. For any two colors α, β, the subgraph generated by the union of α and βcolor classes is a spanning subgraph of which is the union of some cycle subgraphs C 1,..., C s of with disjoint vertices and even length. If has an even 2factor then contains a spanning subgraph s i=1 where are disjoint cycles of even length. Thus are 1factorizable and by Lemma 2.4, there exists a random numbering φ Ci such that for any edge ab of we have φ a (b) = φ b (a). Now define φ x for each x V() as follows: φ x (y) = { φ x (y) xy E( ) 3 otherwise. It is easy to see that φ is a random numbering of satisfying the conditions of Lemma 2.4. (i) (iii) We know is 1factorizable if only if there exists a 1factorizable numbering φ such that for any xy E() we have φ x (y) = φy (x). Assume is 1factorizable and φ is a 1factorizable numbering. If V(C 3 ) = [3] = {1, 2, 3} then we define φ (x,i) R φ C 3 for (x, i) V(R φ C 3 ) as follows: { φ (x,i) [3]{i,j} x = y, ij E(C3 ) R φ C 3 ((y, j)) = φ x (y) xy E(), where (y, j) N Rφ C 3 ((x, i)). It is obvious that φ (x,i) ((y, j)) = φ(y,j) ((x, i)). Now Lemma 2.4 completes the RH RH proof. Corollary 2.6. Let be a 1factorizable graph and φ is 1factorizable numbering of. Assume also that 1 denotes R φ C 3 and recursively let r = r 1 R φ r 1 C 3. Then { r } r 1 is an infinite family of 1factorizable cubic graphs. Theorem 2.7. Let be an (n, k)graph and let H be a (k, k )graph with V(H) = [k]. If H has a 1factorization, then the replacement product R φ H is 1factorizable for any random numbering φ of. Proof. Let RH = R φ H. By Lemma 2.4, there exists a set φ H = {φ a H : N H(a) [k ] a V(H)} such that for any ab E(H) we have φ a H (b) = φb (a). Now define φ(x,a) H RH for any (x, a) V(RH) and (y, b) N RH(x, a) as follows: φ (x,a) { φ RH ((y, b)) = a H (b) k + 1 x = y otherwise. It is obvious that φ (x,a) ((y, b)) = φ(y,b) ((x, a)). Now Lemma 2.4 completes the proof. RH RH Example 2.8. Let C n be a cycle of length n. Then for any random numbering φ Cn, we have C n R φcn K 2 C 2n If n is odd then C n is not 1factorizable but C n R φcn K 2 C 2n is 1factorizable. If n is even then C n 1 is not 1 factorizable. Therefore for any random numbering φ Kn, the graph K n R φkn C n 1 is cubic having an even 2factor. Then by Lemma 2.5, K n R φkn C n 1 is a 1factorizable graph. Theorem 2.9. Let be an (n, m)graph. Then the replacement product R φ K m is 1factorizable whenever one of the following conditions holds: (a) m is even and φ is any random numbering of. (b) is 1factorizable and φ is any 1factorizable numbering of.
5 A. Abdollahi, A. Loghman / Filomat 27:1 (2013), Proof. The complete graph K m is 1factorizable whenever m is even. Then by Theorem 2.7, R φ K m is 1factorizable for any random numbering φ of. If m is odd and is 1factorizable then by Lemma 2.4, there exists a random numbering φ such that for x V() and y N (x), we have φ x (y) = φy (x). For any (x, i) V(R φ K m ) and (y, j) N Rφ K m ((x, i)), where i, j V(K m ) = [m], we define φ (x,i) R φ K m as follows: φ x (y) xy E() φ (x,i) R φ K m ((y, j)) = i+j 2 x = y and i + j is non zero and even m x = y and i + j = 0 i+j+m 2 x = y and i + j is odd where i + j is the remainder when i + j is divided by m. It is easy to see that φ (x,i) RK m ((y, j)) = φ (y,j) RK m ((x, i)). This completes the proof. For the replacement product RC k, k 4, we derive the following result. Theorem Let be an (n, k)graph and C k a cycle of length k 4. Then the replacement product R φ C k is 1factorizable if one of the following conditions holds: (a) k is even and φ is any random numbering of (b) has an even 2factor H = s i=1 and φ Ci = φ Ci, where φ Ci are 1factorizable numbering of. Proof. If k is even then C k = ( k) has 1factorization and by Theorem 2.7, R φ C k is 1factorizable. Let have an even 2factor. Thus there exist disjoint cycles (1 i s) with even lengths, whose s i=1 is a spanning subgraph of the graph. Cycles = (a i1 a i2... a ini ), 1 i s, are 1factorizable and n = s i=1 n i. Because each n i is even then by Lemma 2.4, there are φ Ci where for any edge a il, a im of we have φ a il (a im ) = φ a im (a il ). Let φ a i1 (a i2 ) = φ a i2 (a i1 ) = 1 thus we define cycle T i as follows: T i = ((a i1, 1)(a i2, 1)(a i2, k)(a i2, k 1)... (a i2, 3)(a i2, 2)(a i3, 2)(a i3, 3)... (a i3, k 1)(a i3, k)(a i3, 1)... (a ini, 1)(a ini, k)(a ini, k 1)... (a ini, 3)(a ini, 2)(a i1, 2)(a i1, 3)... (a i1, k 1)(a i1, k)) T i are cycles of length kn i and s i=1 T i is a spanning subgraph of the replacement product R φ C k, where φ Ci = φ Ci. Then by Lemma 2.5, R φ C k has 1factorization., Figure 3: (A) is Icosahedron and (B) is Fullerene C 60
6 A. Abdollahi, A. Loghman / Filomat 27:1 (2013), Corollary Let be an (n, k)graph and 2k n. Then by Dirac s Theorem, the replacement product RC k is 1factorizable. Let us end the paper with two applications of the above result. Figure 4: Coloring the edges of C 60. Figure 5: A TC 4 C 8 (R) tori (a) Top view (b) Side view Example A fullerene graph (in short a fullerene) is a 3connected cubic planar graph, all of whose faces are pentagons and hexagons. By Euler formula the number of pentagons equals 12. The first fullerenes, C 60 and C 70, were isolated in The smaller version, C 60, is in the shape of a soccer ball. raphtheoretic observations on structural properties of fullerenes are important in this respect [3, 5, 6]. Suppose F is a fullerene C 60 and A is the icosahedron ((12, 5)graph). By Figure 3(A), C = (a 1 a 3 a 2 a 6 a 5 a 10 a 11 a 7 a 12 a 8 a 9 a 4 ) is a Hamiltonian cycle of A and A is hamiltonian. If φ A for any edges a i a j are defined in Table I, then AR φa C 5 = F is Hamiltonian, see Figure 3(B). By Lemma 2.5, F is 1factorizable. There are many different ways to color the edges of the graph C 60 (for example see Figure 4). Example Consider C 4 nanotorus S. Then S = C k C m, where C n is the cycle of order n and S be an (km, 4) graph ( S is the Cartesian product of cycles). Suppose T is a T = TC 4 C 8 [k, m](r) nanotorus (which k is the number of squares in every row and m is the number of squares in every column, Figure 5), [2]. Then T is an (4km, 3)graph. It is easy to see that T = SR φs C 4, for some of random numbering φ S of graph S. Because C 4 is 1factorizable then by Theorem 2.10, nanotorus T is 1factorizable.
7 A. Abdollahi, A. Loghman / Filomat 27:1 (2013), φ a 1 A (a 4) φ a 1 A (a 3) φ a 1 A (a 2) φ a 1 A (a 6) φ a 1 A (a 5) φ a 4 A (a 10) φ a 4 A (a 9) φ a 4 A (a 3) φ a 4 A (a 1) φ a 4 A (a 5) φ a 3 A (a 9) φ a 3 A (a 8) φ a 3 A (a 2) φ a 3 A (a 1) φ a 3 A (a 4) φ a 2 A (a 8) φ a 2 A (a 7) φ a 2 A (a 6) φ a 2 A (a 1) φ a 2 A (a 3) φ a 6 A (a 7) φ a 6 A (a 11) φ a 6 A (a 5) φ a 6 A (a 1) φ a 6 A (a 2) φ a 5 A (a 11) φ a 5 A (a 10) φ a 5 A (a 4) φ a 5 A (a 1) φ a 5 A (a 6) φ a 12 A (a 11) φ a 12 A (a 7) φ a 12 A (a 8) φ a 12 A (a 9) φ a 12 φ a 11 A (a 6) φ a 11 A (a 6) φ a 11 A (a 7) φ a 11 A (a 12) φ a 11 φ a 7 A (a 5) φ a 7 A (a 2) φ a 7 A (a 8) φ a 7 A (a 12) φ a 7 A (a 11) φ a 8 A (a 4) φ a 8 A (a 3) φ a 8 A (a 9) φ a 8 A (a 12) φ a 8 A (a 7) φ a 9 A (a 3) φ a 9 A (a 4) φ a 9 A (a 10) φ a 9 A (a 12) φ a 9 A (a 8) φ a 10 A (a 2) φ a 10 A (a 5) φ a 10 A (a 11) φ a 10 A (a 12) φ a 10 A (a 9) A (a 10) A (a 10) Table 1: Table I Acknowledgement. The authors are grateful to the referee for his/her invaluable comments improving the original version of the paper. References [1] N. Alon, A. Lubotzky, and A. Wigderson, Semidirect product in groups and zigzag product in graphs: connections and applications (extended abstract). In 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001), pages IEEE Computer Society, [2] A. R. Ashrafi and A. Loghman, Computing PadmakarIvan index of TUC 4 C 8 (S) Nanotorus, J. Comput. Theor. Nanosci. 3(3) (2006) [3]. Brinkmann, B.D. McKay, Construction of planar triangulations with minimum degree 5, Discrete Math. 301 (2005) [4].A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc., 2(1952), [5] J.E. raver, The independence numbers of fullerenes and benzenoids, European J. Combin. 27 (2006) [6] J.E. raver, Encoding fullerenes and geodesic domes, SIAM. J. Discrete Math. 17 (2004) [7] C. A. Kelley, D. Sridhara and J. Rosenthal, ZigZag and Replacement product graphs and LDPC codes, Advances in Mathematics of Communications, 2(4)(2008) [8] O. Reingold, S. Vadhan and A. Wigderson, Entropy waves, the zigzag graph product, and new constantdegree expanders, Ann. of Math. 2,(155) (2002), [9] V.. Vizing, On an estimate of the chromatic index of a pgraph, Metody Diskret Analiz, 3(1964),
Tools 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 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 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 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 informationA simple proof for the number of tilings of quartered Aztec diamonds
A simple proof for the number of tilings of quartered Aztec diamonds arxiv:1309.6720v1 [math.co] 26 Sep 2013 Tri Lai Department of Mathematics Indiana University Bloomington, IN 47405 tmlai@indiana.edu
More informationZachary Monaco Georgia College Olympic Coloring: Go For The Gold
Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various
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 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 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 informationLabeling 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
More informationPlanarity 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?
More informationStrong Ramsey Games: Drawing on an infinite board
Strong Ramsey Games: Drawing on an infinite board Dan Hefetz Christopher Kusch Lothar Narins Alexey Pokrovskiy Clément Requilé Amir Sarid arxiv:1605.05443v2 [math.co] 25 May 2016 May 26, 2016 Abstract
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 informationCOUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS
COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics
More informationHOMEWORK #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
More informationColoring 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@unilj.si Abstract A simple characterization
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 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 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 informationOn parsimonious edgecolouring of graphs with maximum degree three
On parsimonious edgecolouring of graphs with maximum degree three JeanLuc Fouquet, JeanMarie Vanherpe To cite this version: JeanLuc Fouquet, JeanMarie Vanherpe. On parsimonious edgecolouring of graphs
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 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 informationEdge looseness of plane graphs
Also available at http://amcjournal.eu ISSN 1855966 (printed edn.), ISSN 1855974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (015) 89 96 Edge looseness of plane graphs Július Czap Department of
More informationNotes on Matrix Multiplication and the Transitive Closure
ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.
More informationDiscrete Mathematics & Mathematical Reasoning Chapter 10: Graphs
Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationOn Integer Additive SetIndexers of Graphs
On Integer Additive SetIndexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A setindexer of a graph G is an injective setvalued function f : V (G) 2 X such that
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 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 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 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 informationPROBLEM SET 7: PIGEON HOLE PRINCIPLE
PROBLEM SET 7: PIGEON HOLE PRINCIPLE The pigeonhole principle is the following observation: Theorem. Suppose that > kn marbles are distributed over n jars, then one jar will contain at least k + marbles.
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 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 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 informationTenacity 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
More informationSEQUENCES 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
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 informationZeroSum Magic Labelings and Null Sets of Regular Graphs
ZeroSum Magic Labelings and Null Sets of Regular Graphs Saieed Akbari a,c Farhad Rahmati b Sanaz Zare b,c a Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran b Department
More informationOn 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
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 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 information2. Let H and K be subgroups of a group G. Show that H K G if and only if H K or K H.
Math 307 Abstract Algebra Sample final examination questions with solutions 1. Suppose that H is a proper subgroup of Z under addition and H contains 18, 30 and 40, Determine H. Solution. Since gcd(18,
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 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 informationIntersection Dimension and Maximum Degree
Intersection Dimension and Maximum Degree N.R. Aravind and C.R. Subramanian The Institute of Mathematical Sciences, Taramani, Chennai  600 113, India. email: {nraravind,crs}@imsc.res.in Abstract We show
More informationBasic 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
More informationRamsey numbers for bipartite graphs with small bandwidth
Ramsey numbers for bipartite graphs with small bandwidth Guilherme O. Mota 1,, Gábor N. Sárközy 2,, Mathias Schacht 3,, and Anusch Taraz 4, 1 Instituto de Matemática e Estatística, Universidade de São
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 informationClass One: Degree Sequences
Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of
More informationDiscrete Mathematics Problems
Discrete Mathematics Problems William F. Klostermeyer School of Computing University of North Florida Jacksonville, FL 32224 Email: wkloster@unf.edu Contents 0 Preface 3 1 Logic 5 1.1 Basics...............................
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 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 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 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 informationPlanar Graph and Trees
Dr. Nahid Sultana December 16, 2012 Tree Spanning Trees Minimum Spanning Trees Maps and Regions Eulers Formula Nonplanar graph Dual Maps and the Four Color Theorem Tree Spanning Trees Minimum Spanning
More informationBrendan D. McKay Computer Science Dept., Vanderbilt University, Nashville, Tennessee 37235
139. Proc. 3rd Car. Conf. Comb. & Comp. pp. 139143 SPANNING TREES IN RANDOM REGULAR GRAPHS Brendan D. McKay Computer Science Dept., Vanderbilt University, Nashville, Tennessee 37235 Let n~ < n2
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationLong questions answer Advanced Mathematics for Computer Application If P= , find BT. 19. If B = 1 0, find 2B and 3B.
Unit1: Matrix Algebra Short questions answer 1. What is Matrix? 2. Define the following terms : a) Elements matrix b) Row matrix c) Column matrix d) Diagonal matrix e) Scalar matrix f) Unit matrix OR
More informationON SUPER (a, d)edge ANTIMAGIC TOTAL LABELING OF CERTAIN FAMILIES OF GRAPHS
Discussiones Mathematicae Graph Theory 3 (01) 535 543 doi:10.7151/dmgt.163 ON SUPER (a, d)edge ANTIMAGIC TOTAL LABELING OF CERTAIN FAMILIES OF GRAPHS P. Roushini Leely Pushpam Department of Mathematics
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 informationGeneralized Induced Factor Problems
Egerváry Research Group on Combinatorial Optimization Technical reports TR200207. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.
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 informationCommunicated by Bijan Taeri. 1. Introduction
Transactions on Combinatorics ISSN (print: 518657, ISSN (online: 518665 Vol. 3 No. 3 (01, pp. 39. c 01 University of Isfahan www.combinatorics.ir www.ui.ac.ir STUDYING THE CORONA PRODUCT OF GRAPHS
More informationAvailable 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
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 informationMATHEMATICAL ENGINEERING TECHNICAL REPORTS. An Improved Approximation Algorithm for the Traveling Tournament Problem
MATHEMATICAL ENGINEERING TECHNICAL REPORTS An Improved Approximation Algorithm for the Traveling Tournament Problem Daisuke YAMAGUCHI, Shinji IMAHORI, Ryuhei MIYASHIRO, Tomomi MATSUI METR 2009 42 September
More informationBest Monotone Degree Bounds for Various Graph Parameters
Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer
More informationDegree sequences forcing Hamilton cycles in directed graphs
Degree sequences forcing Hamilton cycles in directed graphs Daniela Kühn 1,2 School of Mathematics University of Birmingham Birmingham, UK Deryk Osthus & Andrew Treglown 1,3 School of Mathematics University
More informationFinding and counting given length cycles
Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected
More informationHigh degree graphs contain largestar factors
High degree graphs contain largestar factors Dedicated to László Lovász, for his 60th birthday Noga Alon Nicholas Wormald Abstract We show that any finite simple graph with minimum degree d contains a
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 informationFall 2015 Midterm 1 24/09/15 Time Limit: 80 Minutes
Math 340 Fall 2015 Midterm 1 24/09/15 Time Limit: 80 Minutes Name (Print): This exam contains 6 pages (including this cover page) and 5 problems. Enter all requested information on the top of this page,
More information= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
More informationThe Open University s repository of research publications and other research outputs
Open Research Online The Open University s repository of research publications and other research outputs The degreediameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:
More informationIntroduction to Graph Theory
Introduction to Graph Theory Allen Dickson October 2006 1 The Königsberg Bridge Problem The city of Königsberg was located on the Pregel river in Prussia. The river divided the city into four separate
More information(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
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 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 informationForests and Trees: A forest is a graph with no cycles, a tree is a connected forest.
2 Trees What is a tree? Forests and Trees: A forest is a graph with no cycles, a tree is a connected forest. Theorem 2.1 If G is a forest, then comp(g) = V (G) E(G). Proof: We proceed by induction on E(G).
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More information3. Markov property. Markov property for MRFs. HammersleyClifford theorem. Markov property for Bayesian networks. Imap, Pmap, and chordal graphs
Markov property 33. Markov property Markov property for MRFs HammersleyClifford theorem Markov property for ayesian networks Imap, Pmap, and chordal graphs Markov Chain X Y Z X = Z Y µ(x, Y, Z) = f(x,
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 informationCOMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS
Bull Austral Math Soc 77 (2008), 31 36 doi: 101017/S0004972708000038 COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V EROVENKO and B SURY (Received 12 April 2007) Abstract We compute
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 informationOn an antiramsey type result
On an antiramsey type result Noga Alon, Hanno Lefmann and Vojtĕch Rödl Abstract We consider antiramsey type results. For a given coloring of the kelement subsets of an nelement set X, where two kelement
More information. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9
Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a
More informationFACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS
International Electronic Journal of Algebra Volume 6 (2009) 95106 FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS Sándor Szabó Received: 11 November 2008; Revised: 13 March 2009
More informationComplete graphs whose topological symmetry groups are polyhedral
Algebraic & Geometric Topology 11 (2011) 1405 1433 1405 Complete graphs whose topological symmetry groups are polyhedral ERICA FLAPAN BLAKE MELLOR RAMIN NAIMI We determine for which m the complete graph
More informationChapter 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.
More informationUniform Multicommodity Flow through the Complete Graph with Random Edgecapacities
Combinatorics, Probability and Computing (2005) 00, 000 000. c 2005 Cambridge University Press DOI: 10.1017/S0000000000000000 Printed in the United Kingdom Uniform Multicommodity Flow through the Complete
More informationChapter 6: Colouring Graphs
17 Coloring Vertices Chapter 6: Colouring Graphs 1. Sections 1719: Qualitative (Can we color a graph with given colors?) Section 20: Quantitative (How many ways can the coloring be done?) 2. Definitions:
More informationRough Semi Prime Ideals and Rough BiIdeals in Rings
Int Jr. of Mathematical Sciences & Applications Vol. 4, No.1, JanuaryJune 2014 Copyright Mind Reader Publications ISSN No: 22309888 www.journalshub.com Rough Semi Prime Ideals and Rough BiIdeals in
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 informationSplit Nonthreshold Laplacian Integral Graphs
Split Nonthreshold Laplacian Integral Graphs Stephen Kirkland University of Regina, Canada kirkland@math.uregina.ca Maria Aguieiras Alvarez de Freitas Federal University of Rio de Janeiro, Brazil maguieiras@im.ufrj.br
More informationA 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
More informationA Study of Sufficient Conditions for Hamiltonian Cycles
DeLeon 1 A Study of Sufficient Conditions for Hamiltonian Cycles Melissa DeLeon Department of Mathematics and Computer Science Seton Hall University South Orange, New Jersey 07079, U.S.A. ABSTRACT A graph
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 informationDISTANCE MAGIC CARTESIAN PRODUCTS OF GRAPHS
1 2 Discussiones Mathematicae Graph Theory xx (xxxx) 1 9 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 DISTANCE MAGIC CARTESIAN PRODUCTS OF GRAPHS Christian Barrientos Department of Mathematics,
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 information