Dynamic Coloring on Restricted Graph Classes.

A proper k-coloring of a graph is an assignment of colors from the set $$\{1,2,\ldots ,k\}$$ to the vertices of the graph such that no two adjacent vertices receive the same color. Given a graph $$G=(V,E)$$ , the Dynamic Coloring problem asks to find a proper k-coloring of G such that for every vertex $$v\in V(G)$$ of degree at least two, there exists at least two distinct colors appearing in the neighborhood of v. The minimum integer k such that there is a dynamic coloring of G using k colors is called the dynamic chromatic number of G and is denoted by $$\chi _d(G)$$ . The problem is NP-complete in general, but solvable in polynomial time on several restricted families of graphs. In this paper, we study the problem on restricted classes of graphs. We show that the problem can be solved in polynomial time on chordal graphs and biconvex bipartite graphs. On the other hand, we show that it is NP-complete on star-convex bipartite graphs, comb-convex bipartite graphs and perfect elimination bipartite graphs. Next, we initiate the study on Dynamic Coloring from the parameterized complexity perspective. First, we show that the problem is fixed-parameter tractable when parameterized by neighborhood diversity or twin-cover. Then, we show that the problem is fixed-parameter tractable when parameterized by the combined parameters clique-width and the number of colors.