Journal of Mathematical Chemistry

, Volume 44, Issue 1, pp 270–285 | Cite as

k-resonant toroidal polyhexes

Oridinal Paper


A toroidal polyhex H(p, q, t) is a cubic bipartite graph embedded on the torus such that each face is a hexagon, which can be described by a string (p, q, t) of three integers (p≥ 1, q≥ 1, 0≤ tp−1). A set \(\mathcal H\) of mutually disjoint hexagons of H(p, q, t) is called a resonant pattern if H(p, q, t) has a prefect matching M such that all haxgons in \(\mathcal H\) are M-alternating. A toroidal polyhex H(p, q, t) is k-resonant if any i (1 ≤ i ≤ k) mutually disjoint hexagons form a resonant pattern. In [16], Shiu, Lam and Zhang characterized 1, 2 and 3-resonant toroidal polyhexes H(p, q, t) for min(p, q)≥ 2. In this paper, we characterize k-resonant toroidal polyhexes H(p, 1, t). Furthermore, we show that a toroidal polyhex H(p, q, t) is k-resonant (k≥ 3) if and only if it is 3-resonant.


Toroidal polyhex Perfect matching Resonant pattern k-resonant 

AMS 2000 Subject Classification

05C10 05C70 05C90 


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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouP.R. China

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