Kekulean

Sheng, R. Rapid Ways to Recognize Kekulean Benzenoid Systems. 153, 211-226 (1990). Schafer, H.-J. Recent Contributions of Kolbe Electrolysis to Organic Synthesis. 152,91-151 (1989). 

Sheng, R. Rapid Ways of Recognize Kekulean Benzenoid Systems. 153, 211-226 (1990). 

Algorithms for Calculating the Number of Kekule Structures and Pauling s Bond Orders in Kekulean Benzenoid Systems... 

The existence of Kekule structures in a benzenoid system is the first fundamental problem in the topological theory of benzenoid systems. It was considered as one of the most difficult open problems in this theory. Many investigations have been made in order to find necessary and sufficient conditions for the existence of Kekule structures in a benzenoid system. Some fairly simple conditions which are both necessary and sufficient have been given in the last few years. In this chapter we review the main results and give a rigorous proof for some necessary and sufficient conditions for the existence of Kekule structures in a benzenoid system. In addition, by using the above results, a construction method of some concealed non-Kekulean benzenoid systems is given. 

A Kekule structure or a 1-factor of a benzenoid system H is an independent edge set in H such that every vertex in H is incident with an edge in the edge set. A benzenoid system is said to be Kekulean if it possesses a Kekule structure, otherwise it is said to be non-Kekulean. It was first pointed out by Clar et al. [1,2] that Kekule structures are of paramount importance for the stability of benzenoid systems. In fact, up to now, no non-Kekulean benzenoid system has been synthesized by chemists. Therefore the existence of Kekule structures in a benzenoid system is a fundamental problem in the topological theory of benzenoid systems. 

After this chemists hoped to find some fairly simple necessary and sufficient conditions. This is why until 1982-1983 I. Gutman and N. Trinajstic still pointed out several times [7-9] that the problem of recognizing Kekulean benzenoid systems was an open problem, and it was thought to be one of the most difficult open problems in the topological theory of benzenoid systems. 

It is well known that a Kekulean benzenoid system must have the same number of black and white vertices, and consequently A = 0. The benzenoid systems with A 4= 0 are said to be obvious non-Kekulean. In the following, we focus our attention on the benzenoid systems with A = = 0. 

The conditions A = 0 is necessary but not sufficient for the existence of Kekule structures. The smallest non-Kekulean benzenoid systems with A = 0 (called concealed non-Kekulean) were found during the years 1972-1986 see, e.g. Bfiinvoll et al. [16], Two of them are shown in Fig. 3. 

If the number of vertices is even, then an evidently sufficient condition for the existence of Kekule structures of a benzenoid system is the existence of a Hamiltonian path [17]. As a corollary, all catacondensed benzenoid systems are Kekulean [18]. But the condition is not necessary. A Kekulean benzenoid system with no Hamilton path is shown in Fig. 4. 

Some other necessary or sufficient (but not necessary and sufficient) criteria for the differentiation between Kekulean and non-Kekulean benzenoid systems can be found in a number of recently published papers [5,19-23], In particular, Sachs gave some important results [5], He introduced the concepts of a cut segment and a cut in a benzenoid system. 

From Theorem 6 and Lemma we see that there exist some concealed non-Kekulean benzenoid systems which satisfy the conditions of the lemma but not (2) of the theorem. We call them the concealed non-Kekulean benzenoid systems of type I. Such an example is shown in Fig. 6. 

It is natural to investigate the smallest concealed non-Kekulean benzenoid systems of type I. It may enable us to find some simpler necessary and sufficient conditions for benzenoid systems with small number of hexagons to have Kekule structures. 

The smallest concealed non-Kekulean benzenoid system of type I was first found by Zhang Fuji and Guo Xiaofeng [27] (see Fig. 9). 

For the benzenoid systems with h < 11, the conditions of Theorem 8 can be further simplified. As stated above, a smallest concealed non-Kekulean benzenoid system has eleven hexagons (see Fig. 3). It implies that, for benzenoid systems with h < 11, (i) of Theorem 8 is also sufficient. By Theorem 8 we can give it a simple proof. 

By computer-aided generation the authors of Ref. [16] found that there are exactly eight smallest concealed non-Kekulean benzenoid systems. In addition, He Wenchen et al. [28] asserted that there are exactly ninety-eight concealed non-Kekulean benzenoid systems with h = 12. 

Using Theorem 8, we can give a construction method for concealed non-Kekulean benzenoid systems with h < 14, which does not depend on computer-aided generation. By this construction method, we proved the above results from References [16 and 28], Furthermore, we have proved that there are exactly 1097 concealed non-Kekulean benzenoid systems with h = 13 [27, 29]. 

We would like to thank Prof. I. Gutman and S. J. Cyvin for their manuscript Kekulean and Non-Kekulean Benzenoid Hydrocarbons prior to publication [30], We would also like to acknowledge support from the NNSFC. 

The present paper is a review of the P-V path method. The concept of the P-V path was proposed by Gordon and Davison in 1952. In the last few years this method has been greatly developed and has become one of the important approaches for investigating Kekule structures of benzenoid hydrocarbons. The superiority of this method is its simplicity and visualization. According to the properties of the P-V path, some algorithmic approaches have been developed for deciding whether a benzenoid or a coronoid hydrocarbon is Kekulean. In 1985, John and Sachs introduced the concept of the P-V matrix and deduced the John-Sachs theorem which states that the absolute value of the determinant of the P-V matrix of a benzenoid or a coronoid hydrocarbon G is equal to the number of Kekule structures of G. 

Only Kekulean benzenoid and coronoid hydrocarbons are known to exist. Non-Kekulean benzenoid and coronoid systems have never been synthesized [2-6], they should be polyradicals and have very low chemical stability. 

Theorem 3. For a Kekulean benzenoid or coronoid system the number of peaks is equal to the number of valleys [11],... 

A = 0 is a necessary but not sufficient condition for Kekulean benzenoids and coronoids. The systems with A 4= 0 are obvious non-Kekulean systems and those non-Kekulean systems with A = 0 are called concealed non-Kekulean systems, (see Fig. 2)... 

Algorithmic Approaches for Deciding Whether a Benzenoid or Coronoid Hydrocarbon is Kekulean or Non-Kekulean... 

A simple method for recognizing an obvious non-Kekulean system is by counting its peaks and valleys. 

But it needs not always be so. A counterexample of the peeling algorithm is shown in Fig. 4 [15, 16], For the Kekulean system in the figure, the P-V path along the extreme left perimeter doesn t belong to any alternating P-V path system of the Kekulean hydrocarbon. 

He conjectured that conversly, if (3) holds for all edge-cuts (and for all positions), then B is Kekulean [8]. 

Theorem 7. Let B be a Kekulean benzenoid system, an edge-cut is made through one of its free edges, e. If t — s > 0, then there exists a Kekule structure of B in which e is a double bond edge. 

Theorems 8. If B is a Kekulean benzenoid system, (u, v) is a convex pair of B, and e is the free edge whose ends are u and v, then B possesses a Kekule structure in which e is a double bond. 

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