Geodesic
K3-WORM Colorings of Graphs: Lower Chromatic Number and Gaps in the Chromatic Spectrum
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 759-772.

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A K3-WORM coloring of a graph G is an assignment of colors to the vertices in such a way that the vertices of each K3-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 K3-WORM-colorable graph in which the minimum number of colors is exactly k. There also exist K3-WORM colorable graphs which have a K3-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.
Keywords: WORM coloring, lower chromatic number, feasible set, gap in the chromatic spectrum 
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Bujtás, Csilla; Tuza, Zsolt. K3-WORM Colorings of Graphs: Lower Chromatic Number and Gaps in the Chromatic Spectrum. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 759-772. http://geodesic.mathdoc.fr/item/DMGT_2016_36_3_a17/

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