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Zero Forcing in Claw-Free Cubic Graphs

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Abstract

In this paper, we study a dynamic coloring of the vertices of a graph G that starts with an initial subset S of colored vertices, with all remaining vertices being uncolored. At each discrete time interval, a colored vertex with exactly one uncolored neighbor forces this uncolored neighbor to be colored. The initial set S is a zero forcing set of G if, by iteratively applying the forcing process, every vertex in G becomes colored. The zero forcing number Z(G) of G is the minimum cardinality of a zero forcing set of G. In this paper, we prove that if G is a connected, cubic, claw-free graph of order \(n \ge 6\), then \(Z(G) \le \alpha (G) + 1\) where \(\alpha (G)\) denotes the independence number of G. Further we prove that if \(n \ge 10\), then \(Z(G) \le \frac{1}{3}n + 1\). Both bounds are shown to be best possible.

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Acknowledgements

The authors express their sincere thanks to the anonymous referees for their very helpful comments and suggestions which greatly improved the exposition and clarity of the revised version of the paper.

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Authors and Affiliations

  1. Department of Pure and Applied Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa

    Randy Davila & Michael A. Henning

  2. Department of Mathematics and Statistics, University of Houston–Downtown, Houston, TX, 77002, USA

    Randy Davila

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  1. Randy Davila
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  2. Michael A. Henning
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Correspondence to Randy Davila.

Additional information

Communicated by Sandi Klavžar.

Michael A. Henning: Research supported in part by the South African National Research Foundation and the University of Johannesburg.

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Davila, R., Henning, M.A. Zero Forcing in Claw-Free Cubic Graphs. Bull. Malays. Math. Sci. Soc. 43, 673–688 (2020). https://doi.org/10.1007/s40840-018-00705-5

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  • DOI: https://doi.org/10.1007/s40840-018-00705-5

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