Three Graphs and the Erd\H{o}s-Gy\'{a}rf\'{a}s Conjecture

Abstract

Three graphs related to the \EGC\, are presented. The graphs are derived from the Buckyball, the Petersen graph, and the Tutte-Coxeter graph. The first graph is a partial answer to a question posed by Heckman and Krakovski \cite{planar} in their recent work on the planar version of the conjecture. The other two graphs appear to be the smallest known cubic graphs with no $2^m$-cycles for $m \leq 4$ and for $m \leq 5$.