Taiwanese Journal of Mathematics

Chromatic Number and Orientations of Graphs and Signed Graphs

Hao Qi, Tsai-Lien Wong, and Xuding Zhu

Full-text: Open access

Abstract

Assume $D$ is a digraph, and $D'$ is a spanning sub-digraph of $D$. We say $D'$ is a modulo-$k$ Eulerian sub-digraph of $D$ if for each vertex $v$ of $D'$, $d_{D'}^+(v) \equiv d_{D'}^-(v) \pmod{k}$. A modulo-$k$ Eulerian sub-digraph $D'$ of $D$ is special if for every vertex $v$, $d_D^+(v) = 0$ implies $d_{D'}^-(v) = 0$ and $d_{D'}^+(v) = d_D^+(v) > 0$ implies $d_{D'}^-(v) > 0$. We denote by $\operatorname{OE}_k(D)$ or $\operatorname{EE}_k(D)$ (respectively, $\operatorname{OE}_k^s(D)$ or $\operatorname{EE}_k^s(D)$) the sets of spanning modulo-$k$ Eulerian sub-digraphs (respectively, the sets of spanning special modulo-$k$ Eulerian sub-digraphs) of $D$ with an odd number or even number of edges. Matiyasevich [A criterion for vertex colorability of a graph stated in terms of edge orientations, (in Russia), Diskretnyi Analiz, issue 26, 65--71 (1974)] proved that a graph $G$ is $k$-colourable if and only if $G$ has an orientation $D$ such that $|\operatorname{OE}_k(D)| \neq |\operatorname{EE}_k(D)|$. In this paper, we give another characterization of $k$-colourable graphs: a graph $G$ is $k$-colourable if and only if $G$ has an orientation $D$ such that $|\operatorname{OE}_{k-1}^s(D)| \neq |\operatorname{EE}_{k-1}^s(D)|$. We extend the characterizations of $k$-colourable graphs to $k$-colourable signed graphs: If $k$ is an even integer, then a signed graph $(G,\sigma)$ is $k$-colourable if and only if $G$ has an orientation $D$ such that $|\operatorname{OE}_k(D)| \neq |\operatorname{EE}_k(D)|$; if $k$ is an odd integer, then $(G,\sigma)$ is $k$-colourable if and only if $G$ has an orientation $D$ such that $|\operatorname{OE}_{k-1}^s(D)| \neq |\operatorname{EE}_{k-1}^s(D)|$, where a (special) modulo-$k$ Eulerian sub-digraph is even or odd if it has an even or odd number of positive edges. The characterization of $k$-colourable signed graphs for even $k$ (respectively, for odd $k$) fails for odd $k$ (respectively, for even $k$).

Article information

Source
Taiwanese J. Math., Volume 23, Number 4 (2019), 767-776.

Dates
Received: 7 February 2018
Revised: 26 August 2018
Accepted: 8 October 2018
First available in Project Euclid: 18 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1563436867

Digital Object Identifier
doi:10.11650/tjm/181005

Mathematical Reviews number (MathSciNet)
MR3982060

Zentralblatt MATH identifier
07088946

Subjects
Primary: 05C15: Coloring of graphs and hypergraphs

Keywords
orientation chromatic number signed graph graph polynomial

Citation

Qi, Hao; Wong, Tsai-Lien; Zhu, Xuding. Chromatic Number and Orientations of Graphs and Signed Graphs. Taiwanese J. Math. 23 (2019), no. 4, 767--776. doi:10.11650/tjm/181005. https://projecteuclid.org/euclid.twjm/1563436867


Export citation

References

  • N. Alon and M. Tarsi, Colorings and orientations of graphs, Combinatorica 12 (1992), no. 2, 125–134.
  • R. C. Brewster, F. Foucaud, P. Hell and R. Naserasr, The complexity of signed graph and edge-coloured graph homomorphisms, Discrete Math. 340 (2017), no. 2, 223–235.
  • T. Gallai, On directed paths and circuits, in: Theory of Graphs (Proc. Colloq., Tihany, 1966), 115–118, Academic Press, New York, 1968.
  • L. A. Goddyn, M. Tarsi and C.-Q. Zhang On $(k,d)$-colorings and fractional nowhere-zero flows, J. Graph Theory 28 (1998), no. 3, 155–161.
  • Y. Kang and E. Steffen, The chromatic spectrum of signed graphs, Discrete Math. 339 (2016), no. 11, 2660–2663.
  • E. Máčajová, A. Raspaud and M. Škoviera, The chromatic number of a signed graph, Electron. J. Combin. 23 (2016), no. 1, Paper 1.14, 10 pp.
  • Yu. V. Matiyasevich, A criterion for vertex colorability of a graph stated in terms of edge orientations, (in Russia), Diskretnyi Analiz, issue 26, 65–71 (1974), Novosibirsk, Institute of Mathematics of Siberian Branch of Academy of Sciences of the USSR.
  • G. J. Minty, A theorem on $n$-coloring the points of a linear graph, Amer. Math. Monthly 67 (1962), no. 7, 623–624.
  • R. Naserasr, E. Rollová and É. Sopena, Homomorphisms of signed graphs, J. Graph Theory 79 (2015), no. 3, 178–212.
  • B. Roy, Nombre chromatique et plus longs chemins d'un graphe, Rev. Française Informat. Recherche Opérationnelle 1 (1967), no. 5, 127–132.
  • Z. Tuza, Graph coloring in linear time, J. Combin. Theory Ser. B 55 (1992), no. 2, 236–243.
  • L. M. Vitaver, Finding minimal vertex coloring of a graph with Boolean powers of the incidence matrix, (in Russian), Dokl. AN SSSR 147 (1962), no. 4, 758–759.
  • T. Zaslavsky, Signed graph coloring, Discrete Math. 39 (1982), no. 2, 215–228.
  • X. Zhu, Circular colouring and orientation of graphs, J. Combin. Theory Ser. B 86 (2002), no. 1, 109–113.