Tight cycle spectrum gaps of cubic 3-connected toroidal graphs

This note complements results on cycle spectra of planar graphs by investigating the toroidal case. Let k≥3 be an integer and G a cubic graph polyhedrally embedded on the torus with circumference at least k. For k=3, it follows from Euler's formula that G has some (facial) cycle of length in [3,6]. For k=4 or 5, we show that G contains a cycle whose length lies in the interval [k,12]; and for k>5, we show that the same holds for the interval [k,2k+4]. This is best possible for all k≥4. On the other hand, for any non-negative integer γ there exist 2-connected cubic graphs G of genus γ, arbitrarily large face-width, and with arbitrarily large gaps in their cycle spectrum. The behaviour of cycle spectrum gaps of graphs polyhedrally embedded on surfaces of genus greater than 1 remains largely unknown.