Minimal reducible bounds for induced-hereditary properties

AbstractLet (Ma,⊆) and (La,⊆) be the lattices of additive induced-hereditary properties of graphs and additive hereditary properties of graphs, respectively. A property R∈Ma (∈La) is called a minimal reducible bound for a property P∈Ma (∈La) if in the interval (P,R) of the lattice Ma (La) there are only irreducible properties. The set of all minimal reducible bounds of a property P∈Ma in the lattice Ma we denote by BM(P). Analogously, the set of all minimal reducible bounds of a property P∈La in La is denoted by BL(P).We establish a method to determine minimal reducible bounds for additive degenerate induced-hereditary (hereditary) properties of graphs. We show that this method can be successfully used to determine already known minimal reducible bounds for k-degenerate graphs and outerplanar graphs in the lattice La. Moreover, in terms of this method we describe the sets of minimal reducible bounds for partial k-trees and the graphs with restricted order of components in La and k-degenerate graphs in Ma

Nontraceable detour graphs

AbstractThe detour order (of a vertex v) of a graph G is the order of a longest path (beginning at v). The detour sequence of G is a sequence consisting of the detour orders of its vertices. A graph is called a detour graph if its detour sequence is constant. The detour deficiency of a graph G is the difference between the order of G and its detour order. Homogeneously traceable graphs are therefore detour graphs with detour deficiency zero. In this paper, we give a number of constructions for detour graphs of all orders greater than 17 and all detour deficiencies greater than zero. These constructions are used to give examples of nontraceable detour graphs with chromatic number k, k⩾2, and girths up to 7. Moreover we show that, for all positive integers l⩾1 and k⩾3, there are nontraceable detour graphs with chromatic number k and detour deficiency l

Some bounds on the generalised total chromatic number of degenerate graphs

The total generalised colourings considered in this paper are colourings of the vertices and of the edges of graphs satisfying the following conditions: • each set of vertices of the graph which receive the same colour induces an m-degenerate graph, • each set of edges of the graph which receive the same colour induces an n-degenerate graph, and • incident elements receive different colours. Bounds for the least number of colours with which this can be done for all k-degenerate graphs are obtained.The first author is thankful to the P.J. Šafárik University, Košice, Slovakia whose hospitality he enjoyed during the preparation of this paper; he is also supported in part by the National Research Foundation of South Africa (Grant Numbers 90841, 91128). The research of the second author was also supported under the grant numbers APVV-15-0091 and VEGA 1/0142/15 and projects ITMS 26220120007 and ITMS 26220220182.http://www.elsevier.com/locate/ipl2018-06-30hb2017Mathematics and Applied Mathematic

Minimum k-path vertex cover

International audienceA 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 P_k(G) the minimum cardinality of a k-path vertex cover in G. It is shown that the problem of determining P_k(G) is NP-hard for each k ≥ 2, while for trees the problem can be solved in linear time. We investigate upper bounds on the value of P_k(G) and provide several estimations and exact values of P_k(G). We also prove that P_3(G) ≤ (2n + m)/6, for every graph G with n vertices and m edges

On some variations of extremal graph problems

A set P of graphs is termed hereditary property if and only if it contains all subgraphs of any graph G belonging to P. A graph is said to be maximal with respect to a hereditary property P (shortly P-maximal) whenever it belongs to P and none of its proper supergraphs of the same order has the property P. A graph is P-extremal if it has a the maximum number of edges among all P-maximal graphs of given order. The number of its edges is denoted by ex(n, P). If the number of edges of a P-maximal graph is minimum, then the graph is called P-saturated and its number of edges is denoted by sat(n, P). In this paper, we consider two famous problems of extremal graph theory. We shall translate them into the language of P-maximal graphs and utilize the properties of the lattice of all hereditary properties in order to establish some general bounds and particular results. Particularly, we shall investigate the behaviour of sat(n,P) and ex(n,P) in some interesting intervals of the mentioned lattice

On generating sets of induced-hereditary properties

A natural generalization of the fundamental graph vertex-colouring problem leads to the class of problems known as generalized or improper colourings. These problems can be very well described in the language of reducible (induced) hereditary properties of graphs. It turned out that a very useful tool for the unique determination of these properties are generating sets. In this paper we focus on the structure of specific generating sets which provide the base for the proof of The Unique Factorization Theorem for induced-hereditary properties of graphs

A note on the Path Kernel Conjecture

AbstractLet τ(G) denote the number of vertices in a longest path in a graph G=(V,E). A subset K of V is called a Pn-kernel of G if τ(G[K])≤n−1 and every vertex v∈V∖K is adjacent to an end-vertex of a path of order n−1 in G[K]. It is known that every graph has a Pn-kernel for every positive integer n≤9. R. Aldred and C. Thomassen in [R.E.L. Aldred, C. Thomassen, Graphs with not all possible path-kernels, Discrete Math. 285 (2004) 297–300] proved that there exists a graph which contains no P364-kernel. In this paper, we generalise this result. We construct a graph with no P155-kernel and for each integer l≥0 we provide a construction of a graph G containing no Pτ(G)−l-kernel

Unique factorization theorem for object-systems

The concept of an object-system is a common generalization of simple graph, digraph and hypergraph. In the theory of generalised colourings of graphs, the Unique Factorization Theorem (UFT) for additive induced-hereditary properties of graphs provides an analogy of the well-known Fundamental Theorem of Arithmetics. The purpose of this paper is to present UFT for object-systems. This result generalises known UFT for additive induced-hereditary and hereditary properties of graphs and digraphs. Formal Concept Analysis is applied in the proof