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Optimal proper connection of graphs

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Abstract

An edge-colored graph G is properly colored if no two adjacent edges share a color in G. An edge-colored connected graph G is properly connected if between every pair of distinct vertices, there exists a path that is properly colored. In this paper, we discuss how to make a connected graph properly connected efficiently. More precisely, we consider the problem to convert a given monochromatic graph into properly connected by recoloring p edges with q colors so that \(p+q\) is as small as possible. We discuss how this can be done efficiently for some restricted graphs, such as trees, complete bipartite graphs and graphs with independence number 2.

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Acknowledgements

The author would like to thank the anonymous referees for careful reading of the manuscript and many helpful comments. This work was supported by JSPS KAKENHI (No. 19K03603).

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  1. School of Data Science, Yokohama City University, 22-2, Seto, Kanazawa-ku, Yokohama, 236-0027, Japan

    Shinya Fujita

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  1. Shinya Fujita
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Correspondence to Shinya Fujita.

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Fujita, S. Optimal proper connection of graphs. Optim Lett 14, 1371–1380 (2020). https://doi.org/10.1007/s11590-019-01442-9

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  • DOI: https://doi.org/10.1007/s11590-019-01442-9

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