Conflict-free coloring on subclasses of perfect graphs and bipartite graphs

Abstract

A Conflict-Free Open Neighborhood coloring, abbreviated CFON^⁎ coloring, of a graph $G=(V,E)$ using k colors is an assignment of colors from a set of k colors to a subset of vertices of V such that every vertex sees some color exactly once in its open neighborhood. The minimum k for which G has a CFON^⁎ coloring using k colors is called the CFON^⁎ chromatic number of G, denoted by ${\chi }_{ON}^{⁎}\left(G\right)$. The analogous notion for closed neighborhood is called CFCN^⁎ coloring and the analogous parameter is denoted by ${\chi }_{CN}^{⁎}\left(G\right)$. The problem of deciding whether a given graph admits a CFON^⁎ (or CFCN^⁎) coloring that uses k colors is NP-complete. Below, we describe briefly the main results of this paper.

• •
  We show that it is NP-hard to determine the CFCN^⁎ chromatic number of chordal graphs. We also show the existence of a family of chordal graphs G that requires $\Omega \left(\sqrt{\omega \left(G\right)}\right)$ colors to CFCN^⁎ color G, where $\omega \left(G\right)$ represents the size of a maximum clique in G.
• •
  We give a polynomial time algorithm to compute ${\chi }_{ON}^{⁎}\left(G\right)$ for interval graphs G. This answers in positive the open question posed by Reddy [Theoretical Comp. Science, 2018] to determine whether CFON^⁎ chromatic number can be computed in polynomial time for interval graphs.
• •
  We explore biconvex graphs, a subclass of bipartite graphs, and give a polynomial time algorithm to compute their CFON^⁎ chromatic number.