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A fully benzenoid system has a unique maximum cardinality resonant set

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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.

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Authors and Affiliations

  1. Faculty of Science, University of Kragujevac, P.O. Box 60, 34000, Kragujevac, Serbia

    Ivan Gutman

  2. Department of Basic Sciences, The British University in Egypt, El Shorouk, 11837, Egypt

    Khaled Salem

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  1. Ivan Gutman
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  2. Khaled Salem
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Correspondence to Khaled Salem.

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Gutman, I., Salem, K. A fully benzenoid system has a unique maximum cardinality resonant set. Acta Appl Math 112, 15–19 (2010). https://doi.org/10.1007/s10440-009-9550-1

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  • DOI: https://doi.org/10.1007/s10440-009-9550-1

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