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

A graph HH is defined to be light in a family HH of graphs if there exists a finite number φ(H,H)φ(H,H) such that each G∈HG∈H which contains HH as a subgraph, contains also a subgraph K≅HK≅H such that the ΔG(K)≤φ(H,H)ΔG(K)≤φ(H,H). We study light graphs in families of polyhedral graphs with prescribed minimum vertex degree δδ, minimum face degree ρρ, minimum edge weight ww and dual edge weight w∗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.