Counterexamples to a conjecture of Merker on 3-connected cubic planar graphs with a large cycle spectrum gap

Abstract

Merker conjectured that if $k\ge 2$ is an integer and $G$ a 3-connected cubic planar graph of circumference at least $k$, then the set of cycle lengths of $G$ must contain at least one element of the interval $[k,2k+2]$. We here prove that for every even integer $k\ge 6$ there is an infinite family of counterexamples.

3 pages, 2 figures