### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 759-772 |

Number of pages | 14 |

Journal | Discussiones Mathematicae - Graph Theory |

Volume | 36 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2016 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

**K3-worm colorings of graphs : Lower chromatic number and gaps in the chromatic spectrum.** / Bujtás, Csilla; Tuza, Z.

Research output: Article

*Discussiones Mathematicae - Graph Theory*, vol. 36, no. 3, pp. 759-772. https://doi.org/10.7151/dmgt.1891

}

TY - JOUR

T1 - K3-worm colorings of graphs

T2 - Lower chromatic number and gaps in the chromatic spectrum

AU - Bujtás, Csilla

AU - Tuza, Z.

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

KW - Feasible set

KW - Gap in the chromatic spectrum

KW - Lower chromatic number

KW - WORM coloring

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UR - http://www.scopus.com/inward/citedby.url?scp=84975742340&partnerID=8YFLogxK

U2 - 10.7151/dmgt.1891

DO - 10.7151/dmgt.1891

M3 - Article

AN - SCOPUS:84975742340

VL - 36

SP - 759

EP - 772

JO - Discussiones Mathematicae - Graph Theory

JF - Discussiones Mathematicae - Graph Theory

SN - 1234-3099

IS - 3

ER -