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 torus with circumference at least . For = , it follows from Euler's formula that has some (facial) length in [ 6 ] 4 or 5, we show contains whose lies interval 12 ; for > 5 same holds 2 + is best possible all On other hand, any non-negative γ there exist 2-connected genus arbitrarily large face-width, gaps their spectrum. The behaviour spectrum surfaces greater than 1 remains largely unknown.