Dynamic Chromatic Number of Bipartite Graphs

Abstract

A dynamic coloring of a graph G is a proper vertex coloring such that for every vertex v ∈ V (G) of degree at least 2, the neighbors of v receive at least 2 colors. The smallest integer k such that G has a dynamic coloring with k colors, is called the dynamic chromatic number of G and denoted by χ 2 (G). Montgomery conjectured that for every r-regular graph G, χ 2 (G) − χ(G) ≤ 2 [19]. Finding an optimal upper bound for χ 2 (G) − χ(G) seems to be an intriguing problem. We show that there is a constant d such that every bipartite graph G with δ(G) ≥ d, has χ 2 (G) − χ(G) ≤ 2⌈ ∆(G) δ(G) ⌉. It was shown that χ 2 (G) − χ(G) ≤ α ′ (G) + k * [2]. Also, χ 2 (G) − χ(G) ≤ α(G) + k * [1]. We prove that if G is a simple graph with δ(G) > 2, then χ 2 (G) − χ(G) ≤ α ′ (G)+ω(G) 2 +k *. Among other results, we prove that for a given bipartite graph G = [X, Y ], determining whether G has a dynamic 4coloring ℓ : V (G) → {a, b, c, d} such that a, b are used for part X and c, d are used for part Y is NP-complete.

$(2/2/3)$-SAT problem and its applications in dominating set problems

“…We say D is a dominating set for T , if for every vertex v ∈ T , we have v ∈ D or there is a vertex u ∈ D such that vu ∈ E(G). For a given graph G and independent set T , finding an independent dominating set D that dominates T has a lot of applications in the concept of dynamic coloring of graphs, see for example [1,2,21]. Motivated by those applications, for an r-regular graph G = (V, E), it was asked in [2] to determine whether for a given independent set T , there is an independent dominating set D that dominates T such that T ∩ D = ∅?…”