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Minimum Number of Edges Guaranteeing the Existence of a \(K_{1, t}\)-Factor in a Graph

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Abstract

Let t,  k,  d be integers with \(1 \le d \le k - 1\) and \(t \ge 8d + 11\), and let \(n = k(t+1)\). We show that if G is a graph of order n such that \(\delta (G) \ge d\) and \(|E(G)| \ge \left( {\begin{array}{c}n\\ 2\end{array}}\right) - (d + 1)n + dt + \frac{1}{2}(d^{2} + 3d + 4)\), then G has a \(K_{1, t}\)-factor. We also give a construction showing the sharpness of the condition on |E(G)|.

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Funding

Funding was supported by Japan Society for the Promotion of Science (17K05347, 20K03720, 19K03603).

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Authors and Affiliations

  1. Applied Mathematics, Faculty of Advanced Science and Technology, Kumamoto University, 2-39-1 Kurokami, Kumamoto, 860-8555, Japan

    Shuya Chiba

  2. Department of Applied Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan

    Yoshimi Egawa

  3. School of Data Science, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama, Kanagawa, 236-0027, Japan

    Shinya Fujita

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  1. Shuya Chiba
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  2. Yoshimi Egawa
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  3. Shinya Fujita
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Correspondence to Shuya Chiba.

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Shuya Chiba was supported by JSPS KAKENHI Grant numbers 17K05347, 20K03720. Shinya Fujita was supported by JSPS KAKENHI Grant number 19K03603.

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Chiba, S., Egawa, Y. & Fujita, S. Minimum Number of Edges Guaranteeing the Existence of a \(K_{1, t}\)-Factor in a Graph. Graphs and Combinatorics 39, 27 (2023). https://doi.org/10.1007/s00373-023-02616-0

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