Number of Kekule Structures as a Function of the Number of Hexagons in Benzenoid Hydrocarbons

The enumeration of Kekule structures has fasci­ nated several researchers since the first systematic studies of benzenoids in terms of graph theory [1-9] , Apart from the recognized importance of Kekule structures in organic and physical chemistry they also have purely mathematical interest. Refer­ ence is made to a recent article in the present jo u r ­ nal [10] along with the bibliography therein. A con­ siderable number of newly published papers on the enumeration of Kekule structures [11-31] shows that the interest in this topic has increased substan­ tially during the last few years. In the present work the term benzenoid is applied in consistence with the definition of the review [32] and the book [33]. The problem of the determ ina­ tion of all benzenoid hydrocarbons with K Kekule structures, 0 < K < 9 was posed several times (see e.g. [34,35]). It has been conjectured that this number is finite [35] and that only one benzenoid has K = 3 (naphthalene), one has K = 4 (anthracene) and two have K = 5 (tetracene and phenanthrene) [35]. In [35] it was demonstrated that K = 2 is obeyed only for the benzene graph. It is known for a long time that there are infinite­ ly many benzenoid systems with nine Kekule struc­ tures. The case K = 9 refers to essentially discon­ nected benzenoids [10], where two naphthalene