A fully benzenoid system has a unique maximum cardinality resonant set
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A fully benzenoid system has a unique maximum cardinality resonant set Ivan Gutman · Khaled Salem
Received: 14 April 2009 / Accepted: 30 September 2009 / Published online: 6 October 2009 © Springer Science+Business Media B.V. 2009
Abstract A benzenoid system is a 2-connected plane graph such that its each inner face is a regular hexagon of side length 1. A benzenoid system is Kekuléan if it has a perfect matching. Let P be a set of hexagons of a Kekuléan benzenoid system B. The set P is called a resonant set of B if the hexagons in P are pair-wise disjoint and the subgraph B − P (obtained by deleting from B the vertices of the hexagons in P ) is either empty or has a perfect matching. It was shown (Gutman in Wiss. Z. Thechn. Hochsch. Ilmenau 29:57– 65, 1983; Zheng and Chen in Graphs Comb. 1:295–298, 1985) that for every maximum cardinality resonant set P of a Kekuléan benzenoid system B, the subgraph B − P is either empty or has a unique perfect matching. A Kekuléan benzenoid system B is said to be fully benzenoid if there exists a maximum cardinality resonant set P of B, such that the subgraph B − P is empty. It is shown that a fully benzenoid system has a unique maximum cardinality resonant set, a well-known statement that, so far, has remained without a rigorous proof. Keywords Benzenoid system · Perfect matching · Resonant set · Clar formula Mathematics Subject Classification (2000) 05C90
1 Introduction A benzenoid system (also called a hexagonal system [1]) is a 2-connected plane graph such that its each inner face is a regular hexagon of side length 1. A benzenoid system is Kekuléan if it has a Kekulé structure, i.e., it has a perfect matching. Benzenoid systems and their perfect matchings were subject to numerous mathematical studies (see, e.g., [1–5]). However,
I. Gutman Faculty of Science, University of Kragujevac, P.O. Box 60, 34000 Kragujevac, Serbia e-mail: [email protected] K. Salem () Department of Basic Sciences, The British University in Egypt, El Shorouk 11837, Egypt e-mail: [email protected]
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I. Gutman, K. Salem
Fig. 1 A resonant set (right) and a maximum cardinality resonant set (left) of coronene
the far greatest interest for benzenoid systems is in chemistry [6–9] since these represent the chemical compounds known as benzenoid hydrocarbons. A necessary condition for a benzenoid hydrocarbon to be (chemically) stable is that it possesses Kekulé structures. Consequently, Kekuléan benzenoid systems pertain to those benzenoid hydrocarbons that are of main chemical interest [10]. The problem of deciding whether a benzenoid system is Kekuléan or not has been much investigated (for review and further references see [11]). In particular, two algorithms with a linear time complexity were designed for this purpose [12, 13]. The present paper reports a result that is related to, and motivated by, the so-called Clar’s aromatic sextet theory. Namely, in the 1970s the German chemist Erich Clar developed a theory of the electronic structure of benzenoid hydrocarbons [14]. For this he used diagrammatica
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