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 $G$, $\delta (H)\le k$. 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 $k\ge 7$, 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.