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

A 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 Δ G ( 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.