Edge-Choosability of Planar Graphs Without Chordal \(7\)-Cycles

Abstract

A graph \(G\) is edge-\(L\)-colorable, if for a given edge assignment \(L = \{L(e) : e \in E(G)\}\), there exists a proper edge-coloring \(\phi\) of \(G\) such that \(\phi(e) \in L(e)\) for all \(e \in E(G)\). If \(G\) is edge-\(L\)-colorable for every edge assignment \(L\) with \(|L(e)| \geq k\) for \(e \in E(G)\), then \(G\) is said to be edge-\(k\)-choosable. In this paper, we prove that if \(G\) is a planar graph without chordal \(7\)-cycles, then \(G\) is edge-\(k\)-choosable, where \(k = \max\{8, \Delta(G) + 1\}\).