Proper Conflict-Free Coloring of Undirected Path, Chordal Bipartite, and Strongly Chordal Graphs

Abstract

A proper conflict-free coloring is a proper coloring in which each nonisolated vertex has a neighbor whose color occurs exactly once in its open neighborhood. We prove that deciding whether the clique number of colors suffices is NP-complete on undirected path graphs, even with a supplied representation by paths in a star. One extra color suffices on every instance in the reduction. We also prove NP-completeness on chordal bipartite graphs for every fixed palette size $k\ge4$. For strongly chordal graphs, reverse simple elimination gives a constructive upper bound of one more than the clique number. We obtain exact polynomial-time algorithms under several conditions on the maximum cliques, including when they induce a connected subtree of a clique tree. For strictly chordal graphs we characterize the optimum and construct an optimal coloring in $O(n+m)$ time, where $n$ and $m$ count the vertices and edges. For strongly chordal graphs with clique-tree separators of size at most $s$, we give an exact algorithm in $n^{2^{O(s+1)}}$ time, with no bound on the clique number.