$K_3$-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum

TL;DR

This paper investigates $K_3$-WORM colorings of graphs, disproves a conjecture about their minimal color count, and explores the computational complexity and special cases of such colorings.

Contribution

It introduces the existence of graphs with prescribed minimal $K_3$-WORM colorings, disproves a conjecture, and analyzes complexity and special graph classes.

Findings

Existence of graphs with minimum $K_3$-WORM coloring exactly $k$ for all $k \\ge 3$

NP-hardness of determining minimum number of colors

Positive results for $d$-degenerate and planar graphs

Abstract

A $K_3$-WORM coloring of a graph $G$ is an assignment of colors to the vertices in such a way that the vertices of each $K_3$-subgraph of $G$ get precisely two colors. We study graphs $G$ which admit at least one such coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014) 161--173] who asked whether every such graph has a $K_3$-WORM coloring with two colors. In fact for every integer $k\ge 3$ there exists a $K_3$-WORM colorable graph in which the minimum number of colors is exactly $k$. There also exist $K_3$-WORM colorable graphs which have a $K_3$-WORM coloring with two colors and also with $k$ colors but no coloring with any of $3,\dots,k-1$ colors. We also prove that it is NP-hard to determine the minimum number of colors and NP-complete to decide $k$-colorability for every $k \ge 2$ (and remains intractable even for graphs of maximum degree 9 if $k=3$). On the other hand, we prove positive results for $d$-degenerate graphs with small $d$, also including planar graphs. Moreover we point out a fundamental connection with the theory of the colorings of mixed hypergraphs. We list many open problems at the end.