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

Tight cycle spectrum gaps of cubic 3-connected toroidal graphs

“…In this case all short faces have length either 3 or 4. It is easy to check that no two short faces of G can intersect as G is a polyhedrally embedded cubic graph with no cycle length in [5,10]. Therefore, we must have G = G 0 , and the last statement follows.…”

“…It is easy to see that this holds for k ∈ {2, 3, 4, 5}. However, in a short note the second author proved that σ 3 0 (k) ≥ 2k + 3 for all even k ≥ 6, see [5]. Cui and the first author [2] expanded on this and gave a full description of the values of σ 0 (k) and σ 3 0 (k) for all k; in particular, they showed that σ 0 (k) = σ 3 0 (k) = 2k + 3 for all k ≥ 10.…”