On large light graphs in families of polyhedral graphs

Abstract

A graph $H$ is said to be light in a family $\mathcal{H}$ of graphs if each graph $G\in \mathcal{H}$ containing a subgraph isomorphic to $H$ contains also an isomorphic copy of $H$ such that each its vertex has the degree (in $G$) bounded above by a finite number $\phi (H,\mathcal{H})$ depending only on $H$ and $\mathcal{H}$. We prove that in the family of all 3-connected plane graphs of minimum degree 5 (or minimum face size 5, respectively), the paths with certain small graphs attached to one of its ends are light.