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The maximum forcing number of cylindrical grid, toroidal 4–8 lattice and Klein bottle 4–8 lattice

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Abstract

Let G be a graph that admits a perfect matching. A forcing set for a perfect matching M of G is a subset S of M, such that S is contained in no other perfect matchings of G. The smallest cardinality of a forcing set of M is called forced matching number, denoted by f(G, M). Among all perfect matchings of G, the maximum forcing matching number is called the maximum forcing number of G, denoted by F(G). In this paper, we show that the maximum forcing numbers of cylindrical grid \(P_{2m}\times C_{2n+1}\) is \(m(n+1)\) by choosing a suitable independent set of this graph. This solves an open problem proposed by Afshani et al. (Australas J Combin 30:147–160, 2004). Moreover, we obtain that the maximum forcing numbers of two classes of toroidal 4–8 lattice and two classes of Klein bottle 4–8 lattice are all equal to the number of squares pq.

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Authors and Affiliations

  1. Department of Mathematics, Huizhou University, Huizhou, 516007, Guangdong, People’s Republic of China

    Xiaoyan Jiang

  2. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People’s Republic of China

    Heping Zhang

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  1. Xiaoyan Jiang
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  2. Heping Zhang
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Correspondence to Xiaoyan Jiang.

Additional information

This work is supported by NSFC (Grant Nos. 11226286 and 11371180).

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Jiang, X., Zhang, H. The maximum forcing number of cylindrical grid, toroidal 4–8 lattice and Klein bottle 4–8 lattice. J Math Chem 54, 18–32 (2016). https://doi.org/10.1007/s10910-015-0541-3

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  • DOI: https://doi.org/10.1007/s10910-015-0541-3

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