On \(3\)-Degeneracy of Some \(C_7\)-free Plane Graphs with Application to Choosability

Abstract

A graph \(G\) is said to be \(k\)-degenerate if for every induced subgraph \(H\) of \(G\), \(\delta(H) \leq k\). Clearly, planar graphs without \(3\)-cycles are \(3\)-degenerate. Recently, it was proved that planar graphs without \(5\)-cycles or without \(6\)-cycles are also \(3\)-degenerate. And for every \(k = 4\) or \(k \geq 7\), there exist planar graphs of minimum degree \(4\) without \(k\)-cycles. In this paper, it is shown that each \(C_7\)-free plane graph in which any \(3\)-cycle is adjacent to at most one triangle is \(3\)-degenerate. So it is \(4\)-choosable.