A complete characterization for \(k\)-resonant Klein-bottle polyhexes (Q926910)

A toroidal polyhex is a 3-regular bipartite graph imbedded in the torus with all faces hexagonal. A hexagonal tessellation \(K(p,q,t)\) on the Klein bottle is a finite-sized elemental benzenoid which can be obtained from a hexagonal lattice on a \(p\times q\)-parallelogram with the usual identification of sides and torsion \(t\). A polyhex is \(k\)-resonant if for each positive integer \(i\) at most \(k\), any \(i\) disjoint hexagons are alternating hexagons with respect to a certain perfect matching. Unlike the toroidal polyhex, \(K(p,q,t)\) is not transitive (except for some degenerate cases). The authors show, however, that \(K(p,q,t)\) is independent of \(t\); they give criteria for \(K(p,q,t)\) to be \(k\)-resonant for every positive integer \(k\), and they show that the \(K(p,q,t)\) of 3-resonance are fully benzenoid.