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Decomposing edge-coloured graphs under colour degree constraints

Part of: Graph theory

Published online by Cambridge University Press:  01 March 2019

Shinya Fujita ,
Ruonan Li  and
Guanghui Wang
Shinya Fujita
Affiliation:
School of Date Science, Yokohama City University, 22-2, Seto, Kanazawa-ku, Yokohama, 236-0027Japan
Ruonan Li
Affiliation:
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, PR China, Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, Xi’an, 710129, PR China and Faculty of EEMCS, University of Twente, PO box 217, 7500 AE Enschede, The Netherlands
Guanghui Wang*
Affiliation:
School of Mathematics, Shandong University, Jinan, 250100, PR China
*
*Corresponding author. Email: ghwang@sdu.edu.cn
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Abstract

For an edge-coloured graph G, the minimum colour degree of G means the minimum number of colours on edges which are incident to each vertex of G. We prove that if G is an edge-coloured graph with minimum colour degree at least 5, then V(G) can be partitioned into two parts such that each part induces a subgraph with minimum colour degree at least 2. We show this theorem by proving amuch stronger form. Moreover, we point out an important relationship between our theorem and Bermond and Thomassen’s conjecture in digraphs.

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 05C20: Directed graphs (digraphs), tournaments
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 5 , September 2019 , pp. 755 - 767
Copyright
© Cambridge University Press 2019 

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Footnotes

†

In this work, an extended abstract for the EUROCOMB 2017 (European Conference on Combinatorics, Graph Theory and Applications, 2017) conference proceedings has been published in [6].

‡

Supported by JSPS KAKENHI (no. 15K04979).

§

Supported by the Fundamental Research Funds for the Central Universities (no. 31020180QD124) and NSFC (no. 11671320, 11671429).

Supported by NSFC (no. 11871193, 11631014).

References

Alon, N. (1997) Disjoint directed cycles. J. Combin. Theory Ser. B 68 167–178.CrossRefGoogle Scholar
Alon, N. (2006) Splitting digraphs. Combin. Probab. Comput. 15 933–937.CrossRefGoogle Scholar
Alon, N., Bang-Jensen, J. and Bessy, S. (2017) Out-colourings of digraphs. arXiv:1706.06441v3Google Scholar
Bermond, J.-C. andThomassen, C. (1981) Cycles in digraphs: A survey. J. Graph Theory 5 1–43.CrossRefGoogle Scholar
Bucić, M. (2017) An improved bound for disjoint directed cycles. Discrete Math. 341 2231–2236.CrossRefGoogle Scholar
Fujita, S., Li, R. andWang, G. (2017) Decomposing edge-colored graphs under color degree constraints. Electron. Notes Discrete Math. 61 491–497.CrossRefGoogle Scholar
Fujita, S., Li, R. and Zhang, S. (2018) Color degree and monochromatic degree conditions for short properly colored cycles. J. Graph Theory 87 362–373.CrossRefGoogle Scholar
Gallai, T. (1967) Transitiv orientierbare Graphen. Acta Math. Hungar. 18 25–66.CrossRefGoogle Scholar
Grossman, J. W. and Häggkvist, R. (1983) Alternating cycles in edge-partitioned graphs. J. Combin. Theory Ser. B 34 77–81.CrossRefGoogle Scholar
Li, R., Broersma, H. and Zhang, S. (2017) Vertex-disjoint properly edge-colored cycles in edge-colored complete graphs. arXiv:1708.08641Google Scholar
Stiebitz, M. (1996) Decomposing graphs under degree constraints. J. Graph Theory 23 321–324.3.0.CO;2-H>CrossRefGoogle Scholar
Stiebitz, M. (2017) A relaxed version of the Erdős–Lovász Tihany conjecture. J. Graph Theory 85 278–287.CrossRefGoogle Scholar
Thomassen, C. (1981) Non-separating cycles in k-connected graphs. J. Graph Theory 5 351–354.CrossRefGoogle Scholar
Thomassen, C. (1983) Disjoint cycles in digraphs. Combinatorica 3 393–396.CrossRefGoogle Scholar
Thomassen, C. (1988) Paths, circuits and subdivisions. In Selected Topics in Graph Theory III (Beineke, L. W. and Wilson, R. J., eds), Academic Press, pp. 97–133.Google Scholar
Thomassen, C. (1989) Configurations in graphs of large minimum degree, connectivity, or chromatic number. In Combinatorial Mathematics: Proceedings of the Third International Conference (New York, 1985), pp. 402–412, New York Academy of Sciences.Google Scholar
Yang, D., Bai, Y., Wang, G. andWu, J. (2018) On splitting digraphs. European J. Combin. 71 174–179.CrossRefGoogle Scholar
Yeo, A. (1997) A note on alternating cycles in edge-colored graphs. J. Combin. Theory Ser. B 69 222–225.CrossRefGoogle Scholar