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

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 ≥ 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. Moreover we point out a fundamental connection with the theory of the colorings of mixed hypergraphs. We list many open problems at the end.2010 Mathematics Subject Classification. 05C15

Coloring subgraphs with restricted amounts of hues

“…That is, if it exists, W .G; P 3 ; f2g/ Ä 2. Now, this result does not generalize to most other F , such as K 3 (see [11]). But we show next that it does generalize somewhat to other stars.…”

“…For example, W .G; K 2 ; f2g/ is just the ordinary chromatic number of G, while several hardness results for WORM and no-rainbow colorings were shown in [5][6][7][8]11]. …”

ℱ-WORM colorings of some 2-trees: partition vectors

“…Bujtás and Zs. Tuza [7,8] also noted this strong relation between F -WORM coloring and mixed hypergraph colorings, a theory that was first introduced by the second author [25,26]. Thus, the notion of F -WORM colorings generalizes several well known coloring constraints.…”

“…Bujtás and Zs. Tuza [5,7,8,12,13]. We note that this coloring requirement makes sense only if each F i ∈ F is of order three or more.…”

Mixed hypergraphs and beyond

“…The notion was introduced not much time ago, in [19], which actually appeared later than the second paper [18]. Further early works on the subject are [11] and [12].…”