Counterexamples to a proposed algorithm for Fries structures of benzenoids

Abstract

The Fries number of a benzenoid is the maximum number of benzenoid hexagons over all of its Kekule structures (perfect matchings), and a Fries canonical structure is a perfect matching that realises this maximum. A recently published algorithm claims to determine Fries canonical structures of benzenoids via iterated Hadamard products based on the adjacency matrix (Ciesielski et al. in Symmetry 2:1390–1400, 2010). This algorithm is re-examined here. Convergence is typically rapid and often yields a single candidate perfect matching, but the algorithm can give an exponential number of choices, of which only a small number are canonical. More worryingly, the algorithm is found to give incorrect results for the Fries number for some benzenoids with as few as seven hexagonal faces. We give a combinatorial reformulation of the algorithm in terms of linear combinations of perfect matchings (with weights at each stage proportional to the products of weights of the edges included in a matching). In all the cases we have examined, the algorithm converges to a maximum-weight matching (or combination of maximum-weight matchings), and where the algorithm fails, either no best Fries matching is of maximum weight, or a best Fries matching is of maximum weight but a sub-optimal matching of the same weight is chosen.