# DOMINATING BIPARTITE SUBGRAPHS IN GRAPHS

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Discussiones Mathematicae Graph Theory 25 (2005 ) DOMINATING BIPARTITE SUBGRAPHS IN GRAPHS Gábor Bacsó Computer and Automation Institute Hungarian Academy of Sciences H 1111 Budapest, Kende u , Hungary Danuta Michalak Faculty of Mathematics Computer Science and Econometrics University of Zielona Góra Podgórna 50, Zielona Góra, Poland and Zsolt Tuza Computer and Automation Institute Hungarian Academy of Sciences and Department of Computer Science University of Veszprém Abstract A graph G is hereditarily dominated by a class D of connected graphs if each connected induced subgraph of G contains a dominating induced subgraph belonging to D. In this paper we characterize graphs hereditarily dominated by classes of complete bipartite graphs, stars, connected bipartite graphs, and complete k-partite graphs. Keywords: dominating set, dominating subgraph, forbidden induced subgraph, bipartite graph, k-partite graph Mathematics Subject Classification: 05C69, 05C38, 05C75.

2 86 G. Bacsó, D. Michalak and Zs. Tuza 1. Introduction The general problem of structural domination can be considered as a subfield of the theory of domination in graphs and also of the theory of induced hereditary properties. It is formulated in [1, 3] in the following way: Basic problem Given a (finite or infinite) class D of connected graphs, characterize the class of those graphs in which every connected induced subgraph contains a dominating induced subgraph isomorphic to some D D. Several papers have been published in which this problem is the focus of study. Researchers have considered various families of connected graphs. The first result of this type can be found in Wolk s paper [7] where D = {K 1 } and the class of hereditarily one-vertex-dominated graphs was characterized in terms of the forbidden induced subgraphs P 4, C 4. The next result (as regards characterization) was given by G. Bacsó and Zs. Tuza [1], and independently by M.B. Cozzens and L.L. Kelleher [5], for dominating cliques. In this case the family of forbidden subgraphs consists of P 5 and C 5. The hereditarily dominated graphs have been characterized for some further families of graphs, too, e.g. for D = {G : diam(g) t} for every given t 2} in [2]. References to various related results, also including sufficient conditions, can be found in [3]. An interesting direction, not really explored so far, was initiated by J. Liu and H. Zhou [6] who characterized the graphs hereditarily dominated by the family of complete bipartite graphs within the class of K 3 -free graphs. This work is the main motivation of our present paper; our Theorem 1 extends its characterization for all graphs, hence dropping the condition of triangle-freeness. 2. Preliminaries We consider only finite, simple graphs. As usual, by V (G) and E(G) we denote the vertex set and the edge set, respectively. Also, K n, P n, C n, and K 1,n denotes the complete graph, the path, and the cycle with n vertices, and the star with n edges, respectively. Moreover, the paw denoted by P W is the graph on vertex set {a, b, c, d} with edge set {ab, ac, ad, bc}. A set D V (G) is called a dominating set if for each v V (G) D there exists a neighbor w D of v; in the other words, N(v) D, where

3 Dominating Bipartite Subgraphs in Graphs 87 N(v) is the set of all vertices in G adjacent to v. A subgraph induced by a dominating set is called a dominating subgraph. Let D be a nonempty class of connected graphs. A graph G is called hereditarily dominated by D if in each of its connected induced subgraphs there exists a dominating induced subgraph belonging to D. A class D of connected graphs is called a compact class if it is closed under taking connected induced subgraphs. Each compact class D is uniquely characterized by a set of graphs minimal not in D, that we shall denote by L(D) = {H / D : each connected, induced, proper subgraph of H is in D}. In the theory of dominating sets, an important role is played by private neighbors, or in other words, private dominated vertices. Let G = (V, E), and D be a dominating induced subgraph of G, with u V (D). We say that u has a private neighbor if there exists a u V (G) \ D such that N(u ) V (D) = {u}. Obviously, if u has a private neighbor, then D \ {u} is not a dominating subgraph in G. In the proofs below, the following concept will occur frequently. Suppose that the graph G is connected, and let H be a subgraph of G. We say that H is d-minimal if it is a connected, dominating induced subgraph, and moreover it is minimal under inclusion with respect to these properties; that is, each of its connected induced subgraphs is non-dominating in G. Being d-minimal implies, in particular, that each non-cutting vertex of H has a private neighbor. To attach a leaf to a given vertex v of G means to take a new vertex v and the edge vv. The leaf-graph of a graph G, denoted F (G), is the graph obtained from G by attaching a leaf to each of its non-cutting vertices. In this paper we consider bipartite graphs and complete k-partite graphs as dominating subgraphs, and obtain characterizations of hereditarily dominated graphs in terms of forbidden induced subgraphs. In each case, the necessity of conditions can be derived immediately from a general method developed in [3], that characterizes the minimally non-d-dominated, non-2- connected graphs in the following way: Cutpoint Lemma [3]. Let D be a compact class. A graph G with at least one cutpoint is minimal non-d-dominated if and only if it is isomorphic to a leaf-graph F (L) where L is a graph minimal not in D.

4 88 G. Bacsó, D. Michalak and Zs. Tuza 3. Dominating Complete Bipartite Subgraphs We begin with considering the class of complete bipartite graphs. Let D 1 = {K p,q : p, q 1} {K 1 }. It is easily checked that the family of graphs minimal not in D 1 is L(D 1 ) = {C 3, P 4 }. Theorem 1. A graph G is hereditarily dominated by D 1 if and only if G is C 6 -free and F (L)-free for all L L(D 1 ). P roof. It is easy to see that none of graphs F (C 3 ), P 6 = F (P 4 ), or C 6 can contain any connected dominating subgraph which is a member of D 1. This proves the only if part. Conversely, let us suppose for a contradiction that there exists a minimal non-d 1 -dominated graph G with no induced subgraph F (C 3 ), P 6, and C 6. Since G is P 6 -free and C 6 -free, each of its d-minimal subgraphs has to be P 4 -free. (This follows, e.g., from the results of [2].) Suppose first that every dominating, connected, induced subgraph of G contains at least one triangle. Among those subgraphs H, we choose one with the minimum possible number of triangles, and furthermore with as many leaves attached to some triangle T H as possible. Let T have t leaves attached inside H. Our goal is to prove t = 3, which is equivalent to saying that F (C 3 ) H. This would lead to the contradiction that G is not F (C 3 )-free, thus some H should be triangle-free. Let x be any vertex of T. If x is a cut-vertex of H, then it is adjacent to some x V (H) which belongs to a connected component of H x not containing T x. This x is then a leaf for T, inside H, attached to x. On the other hand, if x is a non-cutting vertex of H, then the connected subgraph H x contains fewer triangles than H, thus cannot dominate G. We obtain that x has a private neighbor, say x. Then the subgraph H induced by V (H) {x } is connected, dominating, and has precisely the same number of triangles as H does. Inside H, however, the number of leaves attached to T is greater than that in H, i.e., we should have chosen H instead of H. This contradiction proves that t = 3 would indeed hold if H were not triangle-free. Thus, G is dominated by some triangle-free, connected, induced subgraph H. We choose a subgraph H H which is d-minimal. This H remains triangle-free and, by our earlier observations, also P 4 -free. Consequently, H is complete bipartite, contradicting the assumption that G is not D 1 -dominated.

5 Dominating Bipartite Subgraphs in Graphs 89 Using Theorem 1 we can characterize hereditarily dominated graphs for the family of complete bipartite graphs with a bounded number of vertices, too. For n 3, let D 2 = D 2 (n) = {K p,q : 1 p, q n} {K 1 }. The family of minimal graphs not in D 2 is L(D 2 ) = {C 3, P 4, K 1,n+1 }. Theorem 2. A graph G is hereditarily dominated by D 2 (n) if and only if G is C 6 -free and F (L)-free for all L L(D 2 ). P roof. If G contains an induced subgraph F (L) for L L(D 2 ), or an induced C 6, then this subgraph is not dominated by any member of D 2. For the other direction, let us suppose that there exists a minimal non- D 2 -dominated graph G with no induced subgraph F (L), L L(D 2 ) i.e., F (C 3 ), P 6 = F (P 4 ), and F (K 1,n+1 ) and with no induced C 6. Since G does not contain F (C 3 ), P 6 and C 6, applying Theorem 1 we obtain that G is hereditarily dominated by the class of complete bipartite graphs. Hence each induced subgraph of G has a complete bipartite dominating subgraph H. If G is non-d 2 -dominated, then each dominating complete bipartite subgraph H = (V 1, V 2 ) of G has max ( V 1, V 2 ) > n. Say, V 2 n + 1. We choose the dominating H = K p,q so that p is smallest, and with this p the value of q is also smallest. By assumption, we have p 1 and q n Let us choose a vertex a V 1 and denote by b 1,..., b q the vertices in the larger class V 2 of H. By the minimality of q, those vertices have private neighbors, say b 1,..., b q. We consider the possible positions of edges in the subgraph H induced by b 1,..., b q. If H contains a triangle, e.g. b 1, b 2, b 3 are mutually adjacent, then G contains an F (C 3 ) induced by b 1, b 2, b 3, b 1, b 2, b 3. If H contains a (triangle-free) component with more than two vertices, say b 1 b 2 b 3 is an induced P 3, then it forms an induced C 6 together with the vertices b 1, a, b 3. If b 1 b 2 is an isolated edge of H, then these three vertices with b 2, a 1, b 3, b 3 is an induced P 6. Excluding all these possibilities we obtain that the b i are mutually nonadjacent. Thus, the set {a, b 1,..., b q, b 1,..., b q} induces F (K 1,n+1 ), contradicting the assumption that G is F (K 1,n+1 )-free. 4. Dominating Connected Bipartite Subgraphs Next we consider connected bipartite dominating subgraphs. Let D 3 = {G : G is connected and bipartite}. (This also includes K 1.) It is clear that

6 90 G. Bacsó, D. Michalak and Zs. Tuza L(D 3 ) = {C 3, C 5,...}, the set of odd cycles. Theorem 3. A graph G is hereditarily dominated by D 3 if and only if G is F (L)-free for all L L(D 3 ). P roof. The leaf graph F (C 2k+1 ), k 1, does not contain any connected dominating bipartite subgraph, i.e., the condition is necessary. Let us suppose that there exists a minimal non-d 3 -dominated graph G with no induced subgraph F (L) for L L(D 3 ). If G has some cut vertex, then by the Cutpoint Lemma the graph G is isomorphic to F (L) for L L(D 3 ) and the proof is done. Otherwise G x is connected and there exists a dominating subgraph B = B(x) of G x which is a connected bipartite graph, for each vertex x V (G). Adding a neighbor w of x to B, we obtain a dominating subgraph which is almost bipartite. We now apply the method used in [4] for the characterization of graphs hereditarily dominated by paths. We choose the dominating bipartite subgraph B of G x and the neighbor w of x in such a way that the number of induced cycles of odd lengths in H = B {w} is minimum. Let C be an odd cycle in H. We consider the vertices v of C w one by one. If v is a non-cutting vertex of H, then its private neighbor v surely exists, otherwise B v would be a dominating set with fewer odd cycles than B. In this case we insert v into B, hence keeping it bipartite, induced and dominating, with the same number of induced odd cycles. On the other hand, if v is a cut vertex of H, then it has a neighbor v in a component of H v other than the component containing C v. At the end, having found v for each v V (C) and defining w = x as the private neighbor of w, we obtain the contradiction that G contains F (C) as an induced subgraph. (This F (C) is indeed an induced subgraph, for otherwise the extended bipartite graph B with the vertices v would dominate the entire G.) 5. Complete k-partite Dominating Subgraphs Let D 4 = {K n1,n 2,...,n k : k 2, n 1,... n k 1} {K 1 }. The minimal graphs not in D 4 are P 4 and P W, i.e., L(D 4 ) = {P 4, P W }. Theorem 4. A graph G is hereditarily dominated by D 4 if and only if G is C 6 -free and F (L)-free for all L L(D 4 ).

7 Dominating Bipartite Subgraphs in Graphs 91 P roof. It is obvious that the graphs F (P 4 ) = P 6, F (P W ), and C 6 are not dominated by any complete k-partite subgraph. To prove the converse, suppose for a contradiction that G is a minimal counterexample, i.e., G is a minimal non-d 4 -dominated graph that does not contain P 6, C 6, and F (P W ) as induced subgraphs. If G has some cut vertex, then by the Cutpoint Lemma the graph G is isomorphic to F (P W ) or P 6, and the proof is done. Otherwise G x is a connected graph and there exists a dominating subgraph of G x which is a complete k-partite graph, for each x V (G). Since G is P 6 -free and C 6 -free, the d-minimal subgraphs in G contain no induced path P 4. We are going to prove that G is dominated by some P W -free induced subgraph. Suppose not. Let H be a dominating induced subgraph of G, containing an unavoidable paw with vertex set {a, b, c, d} and edge set {ab, ac, ad, bc}, and suppose that the number of triangles inside H is as small as possible. By unavoidable we mean that the leaf vertex d cannot be removed. If d is a non-cutting vertex of H, this assumption means that d has a private neighbor d ; and in the opposite case it is adjacent to some vertex d in a connected component of H d that does not contain a, b, c. Applying now the argument from the proof of Theorem 1, we would obtain leaves b and c for b and c, too. Thus, the contradiction would follow that G is not F (P W )-free. Hence, let H be a P W -free, dominating, connected, induced subgraph of G. Take an induced subgraph of H which is d-minimal in G. By what has been said above, this H is also P 4 -free, so that a complete multipartite dominating subgraph is found. For complete multipartite graphs of bounded-size parts, let n 2 and D 5 = D 5 (n) = {K n1,...,n k : k 2, 1 n 1,..., n k n} {K 1 }. Here the number k of vertex classes is not fixed. The class of graphs minimal not in D 5 is L(D 5 ) = {P 4, P W, K 1,n+1 }. Theorem 5. A graph G is hereditarily dominated by D 5 (n) if and only if G is C 6 -free and F (L)-free for all L L(D 5 ). The proof of Theorem 5 is omitted because it is very similar to that of Theorem 3. We note that an analogous result can be proved also for complete multipartite graphs with a bounded number k of vertex classes. Then K k+1 occurs as a further graph minimal not in D, and hence F (K k+1 ) as a minimal forbidden induced subgraph.

8 92 G. Bacsó, D. Michalak and Zs. Tuza 6. Hereditary Domination by Induced Stars We close this paper with the case of induced stars, which form probably the most interesting subfamily of the complete bipartite graphs. The stars with a restricted number of leaves will also be considered. Let D 6 = {K 1,j : 0 j} and D 7 = D 7 (n) = {K 1,j : 0 j n} for n 3 (where K 1,0 = K 1 ). It is clear that L(D 6 ) = {C 3, C 4, P 4 } and L(D 7 ) = {C 3, C 4, P 4, K 1,n+1 }. Theorem 6. A graph G is hereditarily dominated by D 6 if and only if G is C 6 -free and F (L)-free for all L L(D 6 ). P roof. It is easy to see that an induced C 6 or an induced subgraph F (L) of G, for some L L(D 6 ), does not contain dominating induced stars. Conversely, suppose that there exists a minimal non-d 6 -dominated graph G with no induced subgraph F (C 3 ), P 6 = F (P 4 ) and F (C 4 ), and with no induced C 6. Since G does not contain F (C 3 ), P 6 and C 6, applying Theorem 1 we obtain that G is dominated by some complete bipartite induced subgraphs. We choose a minimal one, say D, with vertex classes A and B. If G is not star-dominated, then A, B 2 and each v A B is a non-cutting vertex of D with at least one private neighbor. We next construct a dominating induced subgraph H, starting from D itself, by sequentially considering the vertices v A B. If v does not have a private neighbor with respect to the subgraph found so far, then we delete it from H unless all elements of A or B would be deleted. And if v still has a private neighbor v, then we insert v into H. At the end of this procedure, subsets A = {a 1,..., a p } A and B = {b 1,..., b q } B remain in H; and if p > 1 and/or q > 1, then every vertex of A and/or B has a private neighbor, mutually nonadjacent. If both p, q 2, then the contradiction F (C 4 ) G is obtained. If p = q = 1, then G is dominated by some path of length l 3. For l 2, a dominating star with at most two leaves is found, contrary to our assumptions; and otherwise for l = 3 the endpoints of this induced dominating path must have private neighbors, hence an induced P 6 or C 6 occurs in G. Finally, assume p = 1 and q 2, and let a 2 A \ A with private neighbor a 2 with respect to D. Since a 2 has been removed from H, a 2 is dominated by some other private neighbor. If it is b 1, then the vertices a 2, b 1, b 1, a 1, b 2, b 2 induce P 6 or C 6. On the other hand, if a 2 is dominated

9 Dominating Bipartite Subgraphs in Graphs 93 by a 1, we check whether H remains dominating after the removal of all the b j. If so, then a dominating star centered at a 1 has been found. And if it isn t, then some b j of H, say b 1 has a private neighbor b 1. Thus, we obtain the final contradiction that a 2, a 1, a 1, b 1, b 1, b 1 induce P 6 or C 6. Theorem 7. A graph G is hereditarily dominated by D 7 (n) if and only if G is C 6 -free and F (L)-free for all L L(D 7 ). P roof. As before, neither the graphs F (L) (L L(D 7 )) nor C 6 contain dominating induced stars on at most n end vertices. Let us suppose that there exists a non-d 7 -dominated graph G with no induced subgraph C 6 and no F (L) for L L(D 7 ); that is, F (C 3 ), P 6, F (C 4 ), F (K 1,n+1 ). Since G does not contain F (C 3 ), F (C 4 ), P 6, and C 6, applying the previous theorem we obtain that G has a dominating induced star H. If G is non-d 7 -dominated, then each minimal dominating star H = K 1,t in G has t n + 1. We notice that, by the minimality of H, each non-cutting vertex of H has a private neighbor. Using the same method as in the proof of Theorem 2, we can easily find F (K n+1 ), and this contradiction completes the proof. Acknowledgements Research of the first and third authors was supported in part by the OTKA Research Fund, grant T References [1] G. Bacsó and Zs. Tuza, Dominating cliques in P 5 -free graphs, Periodica Math. Hungar. 21 (1990) [2] G. Bacsó and Zs. Tuza, Domination properties and induced subgraphs, Discrete Math. 111 (1993) [3] G. Bacsó and Zs. Tuza, Structural domination in graphs, Ars Combin. 63 (2002) [4] G. Bacsó, Zs. Tuza and M. Voigt, Characterization of graphs dominated by paths of bounded length, to appear. [5] M.B. Cozzens and L.L. Kelleher, Dominating cliques in graphs, in: Topics on Domination (R. Laskar and S. Hedetniemi, eds.), Discrete Math. 86 (1990)

10 94 G. Bacsó, D. Michalak and Zs. Tuza [6] J. Liu and H. Zhou, Dominating subgraphs in graphs with some forbidden structures, Discrete Math. 135 (1994) [7] E.S. Wolk, The comparability graph of a tree, Proc. Amer. Math. Soc. 3 (1962) Received 31 October 2003 Revised 16 June 2004

### 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.,

### 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

### The chromatic spectrum of mixed hypergraphs

The chromatic spectrum of mixed hypergraphs Tao Jiang, Dhruv Mubayi, Zsolt Tuza, Vitaly Voloshin, Douglas B. West March 30, 2003 Abstract A mixed hypergraph is a triple H = (X, C, D), where X is the vertex

### Mean Ramsey-Turán numbers

Mean Ramsey-Turán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρ-mean coloring of a graph is a coloring of the edges such that the average

### 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 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, D-98684

### 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.

### 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

### FALSE ALARMS IN FAULT-TOLERANT DOMINATING SETS IN GRAPHS. Mateusz Nikodem

Opuscula Mathematica Vol. 32 No. 4 2012 http://dx.doi.org/10.7494/opmath.2012.32.4.751 FALSE ALARMS IN FAULT-TOLERANT DOMINATING SETS IN GRAPHS Mateusz Nikodem Abstract. We develop the problem of fault-tolerant

### COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

### 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

### 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

### A 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 Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

### 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

### 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

### BOUNDARY EDGE DOMINATION IN GRAPHS

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 0-4874, ISSN (o) 0-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 197-04 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA

### Competition Parameters of a Graph

AKCE J. Graphs. Combin., 4, No. 2 (2007), pp. 183-190 Competition Parameters of a Graph Suk J. Seo Computer Science Department Middle Tennessee State University Murfreesboro, TN 37132, U.S.A E-mail: sseo@mtsu.edu

### 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

### 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

### Fundamentals of Media Theory

Fundamentals of Media Theory ergei Ovchinnikov Mathematics Department an Francisco tate University an Francisco, CA 94132 sergei@sfsu.edu Abstract Media theory is a new branch of discrete applied mathematics

### arxiv:1203.1525v1 [math.co] 7 Mar 2012

Constructing subset partition graphs with strong adjacency and end-point count properties Nicolai Hähnle haehnle@math.tu-berlin.de arxiv:1203.1525v1 [math.co] 7 Mar 2012 March 8, 2012 Abstract Kim defined

### Clique coloring B 1 -EPG graphs

Clique coloring B 1 -EPG graphs Flavia Bonomo a,c, María Pía Mazzoleni b,c, and Maya Stein d a Departamento de Computación, FCEN-UBA, Buenos Aires, Argentina. b Departamento de Matemática, FCE-UNLP, La

### 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

### 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,

### Product irregularity strength of certain graphs

Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (014) 3 9 Product irregularity strength of certain graphs Marcin Anholcer

### 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

### The 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

### 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

### Combinatorial 5/6-approximation of Max Cut in graphs of maximum degree 3

Combinatorial 5/6-approximation of Max Cut in graphs of maximum degree 3 Cristina Bazgan a and Zsolt Tuza b,c,d a LAMSADE, Université Paris-Dauphine, Place du Marechal de Lattre de Tassigny, F-75775 Paris

### Determination of the normalization level of database schemas through equivalence classes of attributes

Computer Science Journal of Moldova, vol.17, no.2(50), 2009 Determination of the normalization level of database schemas through equivalence classes of attributes Cotelea Vitalie Abstract In this paper,

### SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

### Generalized Induced Factor Problems

Egerváry Research Group on Combinatorial Optimization Technical reports TR-2002-07. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.

### PART I. THE REAL NUMBERS

PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS

### 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

### Large 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

### 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

### 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

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### On Some Vertex Degree Based Graph Invariants

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 65 (20) 723-730 ISSN 0340-6253 On Some Vertex Degree Based Graph Invariants Batmend Horoldagva a and Ivan

### 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

### 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

### 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

### DEGREE-ASSOCIATED RECONSTRUCTION PARAMETERS OF COMPLETE MULTIPARTITE GRAPHS AND THEIR COMPLEMENTS

TAIWANESE JOURNAL OF MATHEMATICS Vol. xx, No. xx, pp. 0-0, xx 20xx DOI: 10.11650/tjm.19.2015.4850 This paper is available online at http://journal.taiwanmathsoc.org.tw DEGREE-ASSOCIATED RECONSTRUCTION

### 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.

### Ph.D. Thesis. Judit Nagy-György. Supervisor: Péter Hajnal Associate Professor

Online algorithms for combinatorial problems Ph.D. Thesis by Judit Nagy-György Supervisor: Péter Hajnal Associate Professor Doctoral School in Mathematics and Computer Science University of Szeged Bolyai

### 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

### Degree 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

### Best 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

### THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

### Tree sums and maximal connected I-spaces

Tree sums and maximal connected I-spaces Adam Bartoš drekin@gmail.com Faculty of Mathematics and Physics Charles University in Prague Twelfth Symposium on General Topology Prague, July 2016 Maximal and

### FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS

International Electronic Journal of Algebra Volume 6 (2009) 95-106 FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS Sándor Szabó Received: 11 November 2008; Revised: 13 March 2009

### 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

### 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

### When is a graph planar?

When is a graph planar? Theorem(Euler, 1758) If a plane multigraph G with k components has n vertices, e edges, and f faces, then n e + f = 1 + k. Corollary If G is a simple, planar graph with n(g) 3,

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### 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

### 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

### Tilings of the sphere with right triangles III: the asymptotically obtuse families

Tilings of the sphere with right triangles III: the asymptotically obtuse families Robert J. MacG. Dawson Department of Mathematics and Computing Science Saint Mary s University Halifax, Nova Scotia, Canada

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### 8. 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,

### Ramsey 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

### k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

### 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

### High degree graphs contain large-star factors

High degree graphs contain large-star 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

### ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS

ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS M. AXTELL, J. COYKENDALL, AND J. STICKLES Abstract. We recall several results of zero divisor graphs of commutative rings. We

### On Pebbling Graphs by their Blocks

On Pebbling Graphs by their Blocks Dawn Curtis, Taylor Hines, Glenn Hurlbert, Tatiana Moyer Department of Mathematics and Statistics Arizona State University, Tempe, AZ 85287-1804 November 19, 2008 dawn.curtis@asu.edu

### Triangle 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,

### 2-partition-transitive tournaments

-partition-transitive 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

### Finding 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

### Introduction 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

### 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

### Minimum degree condition forcing complete graph immersion

Minimum degree condition forcing complete graph immersion Matt DeVos Department of Mathematics Simon Fraser University Burnaby, B.C. V5A 1S6 Jacob Fox Department of Mathematics MIT Cambridge, MA 02139

### Lecture 17 : Equivalence and Order Relations DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

### Cycles and clique-minors in expanders

Cycles and clique-minors 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

### A 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

### On the crossing number of K m,n

On the crossing number of K m,n Nagi H. Nahas nnahas@acm.org Submitted: Mar 15, 001; Accepted: Aug 10, 00; Published: Aug 1, 00 MR Subject Classifications: 05C10, 05C5 Abstract The best lower bound known

### Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012

Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it

### On 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

### INDISTINGUISHABILITY 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

### Fundamentele Informatica II

Fundamentele Informatica II Answer to selected exercises 1 John C Martin: Introduction to Languages and the Theory of Computation M.M. Bonsangue (and J. Kleijn) Fall 2011 Let L be a language. It is clear

### Discrete 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

### Graph Theory Problems and Solutions

raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is

### 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

### 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

### ZERO-DIVISOR AND IDEAL-DIVISOR GRAPHS OF COMMUTATIVE RINGS

ZERO-DIVISOR AND IDEAL-DIVISOR GRAPHS OF COMMUTATIVE RINGS T. BRAND, M. JAMESON, M. MCGOWAN, AND J.D. MCKEEL Abstract. For a commutative ring R, we can form the zero-divisor graph Γ(R) or the ideal-divisor

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### Strong 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

### Balloons, Cut-Edges, Matchings, and Total Domination in Regular Graphs of Odd Degree

Balloons, Cut-Edges, Matchings, and Total Domination in Regular Graphs of Odd Degree Suil O, Douglas B. West November 9, 2008; revised June 2, 2009 Abstract A balloon in a graph G is a maximal 2-edge-connected

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### Discrete 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

### A Hajós type result on factoring finite abelian groups by subsets II

Comment.Math.Univ.Carolin. 51,1(2010) 1 8 1 A Hajós type result on factoring finite abelian groups by subsets II Keresztély Corrádi, Sándor Szabó Abstract. It is proved that if a finite abelian group is

### Tree-representation of set families and applications to combinatorial decompositions

Tree-representation of set families and applications to combinatorial decompositions Binh-Minh Bui-Xuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no

### Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

### On the Relationship between Classes P and NP

Journal of Computer Science 8 (7): 1036-1040, 2012 ISSN 1549-3636 2012 Science Publications On the Relationship between Classes P and NP Anatoly D. Plotnikov Department of Computer Systems and Networks,

### Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing

### Stationary random graphs on Z with prescribed iid degrees and finite mean connections

Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the non-negative

### ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER

Séminaire Lotharingien de Combinatoire 53 (2006), Article B53g ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER RON M. ADIN AND YUVAL ROICHMAN Abstract. For a permutation π in the symmetric group

### 1 Introduction. Dr. T. Srinivas Department of Mathematics Kakatiya University Warangal 506009, AP, INDIA tsrinivasku@gmail.com

A New Allgoriitthm for Miiniimum Costt Liinkiing M. Sreenivas Alluri Institute of Management Sciences Hanamkonda 506001, AP, INDIA allurimaster@gmail.com Dr. T. Srinivas Department of Mathematics Kakatiya