A Note on Non-Regular Planar Graphs

Abstract

We give constructive and combinatorial proofs to decide why certain families of slightly irregular graphs have no planar representation and why certain families have such planar representations. Several non-existence results for infinite families as well as for specific graphs are given. For example, the nonexistence of the graphs with $n=11$ and degree sequence $(5,5,5,\dots ,4)$ and $n=13$ and degree sequence $(6,5,5,\dots ,5)$ are shown.