Opuscula Math. 40, no. 4 (2020), 475-482

Opuscula Mathematica

Facial rainbow edge-coloring of simple 3-connected plane graphs

Július Czap

Abstract. A facial rainbow edge-coloring of a plane graph \(G\) is an edge-coloring such that any two edges receive distinct colors if they lie on a common facial path of \(G\). The minimum number of colors used in such a coloring is denoted by \(\text{erb}(G)\). Trivially, \(\text{erb}(G) \geq \text{L}(G)+1\) holds for every plane graph without cut-vertices, where \(\text{L}(G)\) denotes the length of a longest facial path in \(G\). Jendroľ in 2018 proved that every simple \(3\)-connected plane graph admits a facial rainbow edge-coloring with at most \(\text{L}(G)+2\) colors, moreover, this bound is tight for \(\text{L}(G)=3\). He also proved that \(\text{erb}(G) = \text{L}(G)+1\) for \(\text{L}(G)\not\in\{3,4,5\}\). He posed the following conjecture: There is a simple \(3\)-connected plane graph \(G\) with \(\text{L}(G)=4\) and \(\text{erb}(G)=\text{L}(G)+2\). In this note we answer the conjecture in the affirmative.

Keywords: plane graph, facial path, edge-coloring.

Mathematics Subject Classification: 05C10, 05C15.

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  • Július Czap
  • ORCID iD https://orcid.org/0000-0002-9933-5884
  • Technical University of Košice, Faculty of Economics, Department of Applied Mathematics and Business Informatics, Němcovej 32, 040 01 Košice, Slovakia
  • Communicated by Adam Paweł Wojda.
  • Received: 2019-12-27.
  • Revised: .
  • Accepted: 2020-04-26.
  • Published online: 2020-07-09.
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Cite this article as:
Július Czap, Facial rainbow edge-coloring of simple 3-connected plane graphs, Opuscula Math. 40, no. 4 (2020), 475-482, https://doi.org/10.7494/OpMath.2020.40.4.475

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