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.
['Csilla Bujtás', 'Zsolt Tuza']
$K_3$-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum