A Note on the Facial Edge-Coloring Conjecture

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 $$l + 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , 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 $$3l + 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>l</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> colors for any $$l \ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . It is known that the bound $$3l+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>l</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> is tight for general plane graphs. The conjecture was recently confirmed for $$l \le 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> 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 $$l \ge 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , 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 $$\frac{3l + 1}{10}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>l</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>10</mml:mn> </mml:mfrac> </mml:math> .