Abstract
In (J Graph Theory 4:241–242, 1980), Burr proved that \(\chi (G)\le m_1m_2 \ldots m_k\) if and only if G is the edge-disjoint union of k graphs \(G_1,G_2,\ldots ,G_k\) such that \(\chi (G_i)\le m_i\) for \(1\le i\le k\). This result established the practice of describing the chromatic number of a graph G which is the edge-disjoint union of k subgraphs \(G_1,G_2,\ldots ,G_k\) in terms of the chromatic numbers of these subgraphs, and more specific results and conjectures followed. We investigate possible extensions of this theorem of Burr to group coloring and DP-coloring of multigraphs, as well as extensions of another vertex coloring theorem involving arboricity. In particular, we determine the DP-chromatic number of all Halin graphs. In (J Graph Theory 50:123–129, 2005), it is conjectured that for any graph G, the list chromatic number is not higher than the group chromatic number of G. As related results, we show that the group list chromatic number of all multigraphs is at most the DP-chromatic number, and present an example G for which the group chromatic number of G is less than the DP-chromatic number of G.
Similar content being viewed by others
References
Bernshteyn, A.Yu., Kostochka, A.V., Pron, S.P.: On DP-coloring of graphs and multigraphs. Sib. Math. J. 58, 28–36 (2017)
Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, New York (2008)
Burr, S.A.: A Ramsey-theoretic result involving chromatic numbers. J. Graph Theory 4, 241–242 (1980)
Chang, H., Lai, H.-J., Omidi, G.R., Wang, Keke, Zakeri, N.: On group choosability of graphs, II. Graphs Combin. 30, 549–563 (2014)
Chuang, H., Lai, H.-J., Omidi, G.R., Zakeri, N.: On group choosability of graphs. I. Ars Combin. 126, 195–209 (2016)
Dvořák, Z., Postle, L.: Correspondence coloring and its application to list-coloring planar graphs without cycles of lengths 4 to 8. J. Combin. Theory Ser. B 129, 38–54 (2018)
Hungerford, T.W.: Algebra. Springer, New York (2003)
Jaeger, F., Linial, N., Payan, C., Tarsi, M.: Group connectivity of graphs—a non-homogeneous analogue of nowhere-zero flow properties. J. Combin. Theory Ser. B 56, 165–182 (1992)
Jin, L., Wong, T.-L., Zhu, X.: Colouring of generalized signed planar graphs. Discrete Math. 342, 836–843 (2019)
Kim, S.-J., Ozeki, K.: A note on a Brooks’ type theorem for DP-coloring. J. Graph Theory 91, 148–161 (2019)
Král, D., Pangrac, O., Voss, H.-J.: A note on group colorings. J. Graph Theory 50, 123–129 (2005)
Král, D., Nejedlý, P.: Group coloring and list group coloring are $\Pi ^P_2$-complete. In: Lecture Notes in Computer Science, vol. 3153, pp. 274–287. Springer (2004)
Lai, H.-J., Zhang, X.: Group chromatic number of graphs without $K_5$-minors. Graphs Combin. 18, 147–154 (2002)
Lai, H.-J., Zhang, X.: Group colorability of graphs. Ars Combin. 62, 299–317 (2002)
Lai, H.-J., Li, X.: Group chromatic number of graphs. Graphs Combin. 21, 469–474 (2005)
Lai, H.-J., Li, X., Shao, Y.H., Zhan, M.: Group connectivity and group colorings of graphs—a survey. Acta Math. Sin. Engl. Ser. 27, 405–434 (2011)
Lai, H.-J., Omidi, G.R., Raeisi, G.: On group choosability of total graphs. Graphs Combin. 29, 585–597 (2013)
Li, H., Lai, H.-J.: Group colorability of multigraphs. Discrete Math. 313, 101–104 (2013)
Li, X.: Group chromatic number of Halin graphs. Graphs Combin. 31, 1531–1538 (2014)
Funding
The research of Hong-Jian Lai is partially supported by National Natural Science Foundation of China grant (Nos. 11771039, 11771443).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lai, HJ., Mazza, L. Group Colorings and DP-Colorings of Multigraphs Using Edge-Disjoint Decompositions. Graphs and Combinatorics 37, 2227–2243 (2021). https://doi.org/10.1007/s00373-021-02345-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-021-02345-2
Keywords
- Group-coloring
- Group chromatic number
- List group coloring
- Edge-disjoint union of graphs
- DP-coloring
- Correspondence coloring