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

TL;DR

This paper constructs infinite counterexamples to Merker's conjecture, showing that 3-connected cubic planar graphs can have large cycle spectrum gaps, challenging previous assumptions about cycle length distributions.

Contribution

The paper provides the first known infinite family of counterexamples to Merker's conjecture for even integers $k \, \ge \, 6$, disproving the conjecture.

Findings

Counterexamples exist for all even $k \ge 6$

Counterexamples form an infinite family

Disproves Merker's conjecture for these cases

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.