K-3-WORM COLORINGS OF GRAPHS: LOWER CHROMATIC NUMBER AND GAPS IN THE CHROMATIC SPECTRUM

Bujtás, Csilla and Tuza, Zsolt (2016) K-3-WORM COLORINGS OF GRAPHS: LOWER CHROMATIC NUMBER AND GAPS IN THE CHROMATIC SPECTRUM. DISCUSSIONES MATHEMATICAE GRAPH THEORY, 36 (3). pp. 759-772. ISSN 1234-3099

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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] 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.

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