In this paper, we deal with zero-divisor graphs of posets. We prove that the diameter of such a graph is either 1, 2 or 3 while its girth is either 3, 4 or ∞. We also characterize zero-divisor graphs of posets in terms of their diameter and girth. © 2012 Elsevier B.V. All rights reserved.

In the Connected Dominating Set problem we are given as input a graph G and a positive integer k, and are asked if there is a set S of at most k vertices of G such that S is a dominating set of G and the subgraph induced by S is connected. This is a basic connectivity problem that is known to be NP-complete, and it has been extensively studied using several algorithmic approaches. In this paper...

A graph G is called (k, 1)-colorable, if the vertex set of G can be partitioned into subsets V1 and V2 such that the graph G[V1] induced by the vertices of V1 has maximum degree at most k and the graph G[V2] induced by the vertices of V2 has maximum degree at most 1. We prove that every graph with a maximum average degree less than 10k+22 3k+9 admits a (k, 1)-coloring, where k ≥ 2. In particula...

A proper coloring of the vertices of a graph is called a star coloring if the union of every two color classes induce a star forest. The star chromatic number χs(G) is the smallest number of colors required to obtain a star coloring of G. In this paper, we study the relationship between the star chromatic number χs(G) and the maximum average degree Mad(G) of a graph G. We prove that: 1. If G is...

Kostochka, A.V., List edge chromatic number of graphs with large girth, Discrete Mathematics 101 (1992) 189-201. It is shown that the list edge chromatic number of any graph with maximal degree A and girth at least 8A(ln A + 1.1) is equal to A + 1 or to A.

A P -decomposition of a graph G is a set of pairwise edge-disjoint paths of G with edges that cover the edge set of G. Kotzig (1957) proved that a 3-regular graph admits a P3-decomposition if and only if it contains a perfect matching, and also asked what are the necessary and sufficient conditions for an -regular graph to admit a P -decomposition, for odd . Let g, and m be positive integers wi...

Weakly s-arc transitive graphs are introduced and determined. A graph is said to be weakly s-arc transitive if its endomorphism monoid acts transitively on the set of s-arcs. The main results are: (1) A nonbipartite graph is weakly s-arc transitive if and only if it is s-arc transitive. (2) A tree with diameter d is weakly s-arc transitive for all 0 s d. (3) A bipartite graph with girth g = 2s ...

We show that every (sub)cubic n-vertex graph with sufficiently large girth has fractional chromatic number at most 2.2978 which implies that it contains an independent set of size at least 0.4352n. Our bound on the independence number is valid to random cubic graphs as well as it improves existing lower bounds on the maximum cut in cubic graphs with large girth.

We improve some recent results on graph eigenvalues. In particular, we prove that if G is a graph of order n 2; maximum degree ; and girth at least 5; then