K₃-Worm Colorings of Graphs: Lower Chromatic Number and Gaps in the Chromatic Spectrum

• Autorzy: Csilla Bujtás , Zsolt Tuza
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

WORM coloring; lower chromatic number; feasible set; gap in the chromatic spectrum

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2016
• Tom: 36
• Numer: 3
• Strony: 759-772

Daty

wydano

2016-08-01

otrzymano

2015-09-08

poprawiono

2015-11-27

zaakceptowano

2015-11-27

online

2016-07-06

Twórcy

autor

Csilla Bujtás

• Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences, Budapest, Hungary Department of Computer Science and Systems Technology University of Pannonia, Veszprém, Hungary

autor

Zsolt Tuza

• Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences, Budapest, Hungary Department of Computer Science and Systems Technology University of Pannonia, Veszprém, Hungary

Bibliografia

• [1] S. Arumugam, B.K. Jose, Cs. Bujtás and Zs. Tuza, Equality of domination and transversal numbers in hypergraphs, Discrete Appl. Math. 161 (2013) 1859-1867. doi:10.1016/j.dam.2013.02.009[WoS][Crossref]
• [2] G. Bacsó, Cs. Bujtás, Zs. Tuza and V. Voloshin, New challenges in the theory of hypergraph coloring, in: Advances in Discrete Mathematics and Applications, Ramanujan Mathematical Society Lecture Notes Series 13 (2010) 45-57.
• [3] Cs. Bujtás, E. Sampathkumar, Zs. Tuza, Ch. Dominic and L. Pushpalatha, Vertex coloring without large polychromatic stars, Discrete Math. 312 (2012) 2102-2108. doi:10.1016/j.disc.2011.04.013[WoS][Crossref]
• [4] Cs. Bujtás, E. Sampathkumar, Zs. Tuza, M.S. Subramanya and Ch. Dominic, 3- consecutive C-colorings of graphs, Discuss. Math. Graph Theory 30 (2010) 393-405. doi:10.7151/dmgt.1502[Crossref]
• [5] Cs. Bujtás and Zs. Tuza, Maximum number of colors: C-coloring and related problems, J. Geom. 101 (2011) 83-97. doi:10.1007/s00022-011-0082-2[Crossref]
• [6] Cs. Bujtás and Zs. Tuza, Kn-WORM colorings of graphs: Feasible sets and upper chromatic number, manuscript in preparation (2015).
• [7] Cs. Bujtás, Zs. Tuza and V.I. Voloshin, Hypergraph colouring, Chapter 11 in: L.W. Beineke and R.J. Wilson, (Eds.), Topics in Chromatic Graph Theory, Encyclopedia of Mathematics and Its Applications 156 (Cambridge University Press, 2014), 230-254.
• [8] W. Goddard, K. Wash and H. Xu, WORM colorings forbidding cycles or cliques, Congr. Numer. 219 (2014) 161-173.
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• [15] K. Ozeki, private communication, June 2015.
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Identyfikatory

DOI

10.7151/dmgt.1891