Dynamic coloring on restricted graph classes - Bhyravarapu, Sriram; Kumari, Swati; Reddy, I. Vinod (Theoretical computer science, 2025)
A proper k-coloring of a graph is an assignment of colors from the set {1,2,…,k} to the vertices of the graph such that no two adjacent vertices receive the same color. Given a graph G and an integer k, the Dynamic Coloring problem asks whether there exists a proper k-coloring of G such that for every vertex v 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 exists a dynamic coloring of G using k colors is called the dynamic chromatic number of G, denoted by χd(G). The problem is NP-complete in general but is solvable in polynomial time on trees and graphs of bounded tree-width. 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 prove that it is NP-complete on star-convex bipartite graphs, comb-convex bipartite graphs and perfect elimination bipartite graphs. Next, we initiate the study of Dynamic Coloring from the perspective of parameterized complexity. We show that the problem is W[1]-hard when parameterized by clique-width and we present a fixed-parameter tractable algorithm parameterized by the combined parameters clique-width and the number of colors. We also show that the problem is fixed-parameter tractable when parameterized by neighborhood diversity or twin-cover.