Conflict-Free Coloring on Open Neighborhoods

TL;DR

This paper advances the understanding of conflict-free coloring in graphs by providing new upper bounds for planar and outerplanar graphs and analyzing color requirements for Kneser graphs, addressing open questions about minimal color counts.

Contribution

It improves upper bounds for conflict-free coloring of planar and outerplanar graphs and characterizes color requirements for Kneser graphs in specific cases.

Findings

Planar graphs can be conflict-free colored with at most six colors.

Outerplanar graphs can be conflict-free colored with at most four colors.

Kneser graphs require exactly $k+2$ colors when $n extgreater k(k+1)^2 + 1$.

Abstract

In an undirected graph, a conflict-free coloring (with respect to open neighborhoods) is an assignment of colors to the vertices of the graph $G$ such that every vertex in $G$ has a uniquely colored vertex in its open neighborhood. The conflict-free coloring problem asks to find the smallest number of colors required for a conflict-free coloring. The conflict-free coloring problem is NP-complete. From results in Abel et. al. [SODA 2017], it can be inferred that every planar graph has a conflict-free coloring with at most nine colors. As the best known lower bound for planar graphs is four colors, it was asked in the same paper if fewer colors would suffice. We make progress in answering this question, by showing that every planar graph can be colored using at most six colors. The same proof idea is used to show that every outerplanar graph can be colored using at most five colors. Using a different approach, we further show that every outerplanar graph can be colored using at most four colors. Finally, we study the problem on Kneser graphs. We show that $k+2$ colors are necessary and sufficient to color the Kneser graph $K(n,k)$ when $n\geq k(k+1)^2 + 1$.