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.