A Note on the Facial Edge-Coloring Conjecture
Year: 2025
Type: article
Abstract: Let G be a connected plane graph that can have loops and multiple edges. An l-facial edge-coloring of a plane graph G is a coloring of edges of G such that any two edges, that share the same facial trail of length at most $$l + 1$$ l + 1 , receive distinct colors. It is an edge variant of the l-facial vertex coloring, which arose as a generalization of the well-known cyclic coloring. It was conjectured by Lužar et al. in 2015 that every plane graph admits an l-facial edge-coloring with at most $$3l + 1$$ 3 l + 1 colors for any $$l \ge 1$$ l ≥ 1 . It is known that the bound $$3l+1$$ 3 l + 1 is tight for general plane graphs. The conjecture was recently confirmed for $$l \le 3$$ l ≤ 3 by Horňák, Lužar and Štorgel (3-facial edge-coloring of plane graphs, Discrete Math. 346 (2023) 113312). In this note we prove that the conjecture holds, in the case when $$l \ge 4$$ l ≥ 4 , for every graph whose reduction (the graph obtained from G by suppressing all its 2-vertices) is 3-edge connected, and the length of the longest path in G with interior vertices of degree 2 is at most $$\frac{3l + 1}{10}$$ 3 l + 1 10 .
Source: Graphs and Combinatorics
Authors: Stanislav Jendrol’, Alfréd Onderko
Institution: University of Pavol Jozef Šafárik
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,190
OpenAlex ID: W4407946574