Proper Conflict-Free Coloring on Structured Graph Classes: Bounds, Exact Algorithms, and Hardness

Abstract

A proper conflict-free coloring is a proper vertex coloring in which every non-isolated vertex has a neighbor whose color appears exactly once in its open neighborhood. We study this coloring on chordal and ordered bipartite graph classes. For well-partitioned chordal graphs, we prove the tight bound $ω(G)\leq\pcf(G)\leqω(G)+1$ and give an $O(n^2)$ construction from a partition tree. Every assignment of lists of at least $ω(G)+1$ colors also admits a PCF coloring that respects the lists. Deciding whether $ω(G)$ colors suffice is NP-complete, even when the partition tree is a star. This also proves NP-completeness on chordal graphs when the number of colors is part of the input. For interval graphs, we prove that lists of size $ω(G)+1$ always suffice, improving the previous ordinary-coloring bound of $ω(G)+2$. We use matchings to characterize when $ω(G)$ colors suffice and give an endpoint algorithm taking $2^{O(ω(G)^2)}n^{O(1)}$ time. We also characterize the exact value when the clique number is at most three and give a linear-time algorithm from sorted interval endpoints. For biconvex graphs, we prove a tight five-color bound and give a direct exact algorithm taking $O(n^{33})$ time. For bipartite permutation graphs, a smaller dynamic program gives an exact $O(n^{17})$ algorithm. Finally, if a bipartite graph is convex over an ordered shore $X$, we prove the sharp bound $\pcf(G)\leq\lceil\log_2(|X|+1)\rceil+3$ and construct a coloring in linear time from the order.