K3-Worm Colorings of Graphs: Lower Chromatic Number and Gaps in the Chromatic Spectrum

摘要

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] by proving that for every integer k ≥ 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, . . , 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 ≥ 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.