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Facial Rainbow Edge-Coloring of Plane Graphs

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Abstract

An edge-coloring of a loopless plane graph G is a facial rainbow edge-coloring if any two edges of G contained in the same facial path have distinct colors. The facial rainbow edge-number of a graph G, denoted \(\mathrm {erb}(G)\), is the minimum number of colors that are necessary in any facial rainbow edge-coloring. In the present note we prove that \(\mathrm {erb}(G) \le \lfloor \frac{3}{2} (L(G) + 1) \rfloor \) for all connected loopless plane graphs, where L(G) is the length of the longest facial path of G. This bound is tight. For the family of all 3-connected plane graphs this bound is improved to \(L(G) + 2\). For trees there is \(\mathrm {erb}(G) \le \lfloor \frac{3}{2} L(G) \rfloor \) which is also tight. Moreover, if G is a tree with \(L(G) \ge 7\) and without degree two vertices, then \(\mathrm {erb}(G) = L(G)\).

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Acknowledgements

This work was supported by the Slovak VEGA Grant 1/0368/16 and by the Slovak Research and Development Agency under the Contract No. APVV-15-0116.

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  1. Institute of Mathematics, Faculty of Science, P. J. Šafárik University, Jesenná 5, 040 01, Kosice, Slovakia

    Stanislav Jendrol’

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  1. Stanislav Jendrol’
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Correspondence to Stanislav Jendrol’.

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Jendrol’, S. Facial Rainbow Edge-Coloring of Plane Graphs. Graphs and Combinatorics 34, 669–676 (2018). https://doi.org/10.1007/s00373-018-1904-x

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