Marko Jakovac

The b-Chromatic Number of Cubic Graphs

The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-coloring in which every color class contains at least one vertex adjacent to some vertex in all the other color classes. It is proved that with four exceptions, the b-chromatic number of cubic graphs is 4. The exceptions are the Petersen graph, K3,3, the prism over K3, and one more sporadic example on 10 vertices.

ON THE B-CHROMATIC NUMBER OF REGULAR GRAPHS

Abstract The b-chromatic number of a graph G is the largest integer k , such that G admits a proper k -coloring in which every color class contains at least one vertex that has a neighbor in each of the other color classes. We prove that every d -regular graph with at least 2 d 3 vertices has b-chromatic number d + 1 , the b-chromatic number of an arbitrary d -regular graph with girth g = 5 is at least ⌊ d + 1 2 ⌋ and every d -regular graph, d ≥ 6 , with diameter at least d and with no 4-cycles, has b-chromatic number d + 1 .

On the k-path vertex cover of some graph products

A subset S of vertices of a graph G is called a k-path vertex cover if every path of orderk inG contains at least one vertex fromS. Denote by k(G) the minimum cardinality of a k-path vertex cover in G. In this paper improved lower and upper bounds for k of the Cartesian and the direct product of paths are derived. It is shown that for 3 those bounds are tight. For the lexicographic product bounds are presented for k, moreover 2 and 3 are exactly determined for the lexicographic product of two arbitrary graphs. As a consequence the independence and the dissociation number of the lexicographic product are given.

The partition dimension of strong product graphs and Cartesian product graphs

Abstract Let G = ( V , E ) be a connected graph. The distance between two vertices u , v ∈ V , denoted by d ( u , v ) , is the length of a shortest u , v -path in G . The distance between a vertex v ∈ V and a subset P ⊂ V is defined as min { d ( v , x ) : x ∈ P } , and it is denoted by d ( v , P ) . An ordered partition { P 1 , P 2 , … , P t } of vertices of a graph G , is a resolving partition of G , if all the distance vectors ( d ( v , P 1 ) , d ( v , P 2 ) , … , d ( v , P t ) ) are different. The partition dimension of G is the minimum number of sets in any resolving partition of G . In this article we study the partition dimension of strong product graphs and Cartesian product graphs. Specifically, we prove that the partition dimension of the strong product of graphs is bounded below by four and above by the product of the partition dimensions of the factor graphs. Also, we give the exact value of the partition dimension of strong product graphs when one factor is a complete graph and the other one is a path or a cycle. For the case of Cartesian product graphs, we show that its partition dimension is less than or equal to the sum of the partition dimensions of the factor graphs minus one. Moreover, we obtain an upper bound on the partition dimension of Cartesian product graphs, when one factor is a complete graph.

The k -path vertex cover of rooted product graphs

A subset S of vertices of a graph G is called a k -path vertex cover if every path of order k in G contains at least one vertex from S . Denote by ? k ( G ) the minimum cardinality of a k -path vertex cover in G . In this article a lower and an upper bound for ? k of the rooted product graphs are presented. Two characterizations are given when those bounds are attained. Moreover ? 2 and ? 3 are exactly determined. As a consequence the independence and the dissociation number of the rooted product are given.

The b-chromatic number and related topics—A survey

Abstract The b-chromatic number of a graph G is the largest integer k such that G admits a proper k -coloring in which every color class contains at least one vertex that has a neighbor in each of the other color classes. In this survey we present the most important results on b-colorings, b-chromatic number, and related topics.

The security number of strong grid-like graphs

Abstract The concept of a secure set in graphs was first introduced by Brigham et al. in 2007 as a generalization of defensive alliances in graphs. Defensive alliances are related to the defense of a single vertex. However, in a general realistic settings, a defensive alliance should be formed so that any attack on the entire alliance or any subset of the alliance can be defended. In this sense, secure sets represent an attempt to develop a model of this situation. Given a graph G = ( V , E ) and a set of vertices S ⊆ V of G, the set S is a secure set if it can defend every attack of vertices outside of S, according to an appropriate definition of “attack” and “defense”. The minimum cardinality of a secure set in G is the security number s ( G ) . In this article we obtain the security number of grid-like graphs, which are the strong products of paths and cycles (grids, cylinders and toruses). Specifically we show that for any two positive integers m , n ≥ 4 , s ( P m ⊠ P n ) = min ⁡ { m , n , 8 } , s ( P m ⊠ C n ) = min ⁡ { 2 m , n , 16 } and s ( C m ⊠ C n ) = min ⁡ { 2 m , 2 n , 32 } .

The security number of lexicographic products

Abstract A subset S of vertices of a graph G is a secure set if |N [X] ∩ S| ≥ |N [X] − S| holds for any subset X of S, where N [X] denotes the closed neighborhood of X. The minimum cardinality s(G) of a secure set in G is called the security number of G. We investigate the security number of lexicographic product graphs by defining a new concept of tightly-securable graphs. In particular we derive several exact results for different families of graphs which yield some general results.

Ljubljana - Leoben Graph Theory Seminar : Maribor, 13.-15. September, 2017 Book of Abstracts

The booklet contains the abstracts of the talks given at the 30th Ljubljana-Leoben Graph Theory Seminar that was held at the Faculty of Natural Sciences and Mathematics in Maribor between 13-15 September, 2017. The seminar attracted more than 30 participants from eight countries, all of which are researchers in different areas of graph theory. The topics of the talks encompass a wide range of contemporary graph theory research, notably, various types of graph colorings (b-coloring, packing coloring, edge colorings), graph domination (rainbow domination, Grundy domination, graph security), distinguishing problems, algebraic graph theory, graph algorithms, chemical graph theory, coverings, matchings and also some classical extremal problems. Beside the abstracts of the four invited speakers (Csilla Bujtas, Premysl Holub, Jakub Przybylo, Zsolt Tuza), the booklet contains also the abstracts of 18 contributed talks given at the event.