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

AbstractA graph H is defined to be light in a family H of graphs if there exists a finite number φ(H,H) such that each G∈H which contains H as a subgraph, contains also a subgraph K≅H such that the ΔG(K)≤φ(H,H). We study light graphs in families of polyhedral graphs with prescribed minimum vertex degree δ, minimum face degree ρ, minimum edge weight w 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