Light graphs in families of polyhedral graphs with prescribed minimum degree, face size, edge and dual edge weight

摘要

A graph H is defined to be light in a family H of graphs if there exists a finite number phi(H, H) such that each G is an element of H which contains H as a subgraph, contains also a subgraph K congruent to H such that the Delta(G)(K) <= phi(H, H). We study light graphs in families of polyhedral graphs with prescribed minimum vertex degree delta, minimum face degree rho, minimum edge weight in and dual edge weight w*. For those families, we show that there exists a variety of small light cycles; on the other hand, we also present particular constructions showing that, for certain families, the spectrum of short cycles contains irregularly scattered cycles that are not light.