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Proper Colorings of Plane Quadrangulations Without Rainbow Faces

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Abstract

We consider a proper coloring of a plane graph such that no face is rainbow, where a face is rainbow if any two vertices on its boundary have distinct colors. Such a coloring is said to be proper anti-rainbow. A plane quadrangulation G is a plane graph in which all faces are bounded by a cycle of length 4. In this paper, we show that the number of colors in a proper anti-rainbow coloring of a plane quadrangulation G does not exceed \(3\alpha (G)/2\), where \(\alpha (G)\) is the independence number of G. Moreover, if the minimum degree of G is 3 or if G is 3-connected, then this bound can be improved to \(5\alpha (G)/4\) or \(7\alpha (G)/6 + 1/3\), respectively. All of these bounds are tight.

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Acknowledgements

We thank the referees for reading the paper carefully and giving helpful comments and suggestions. In particular, one referee suggested the problem on the existence of a plane quadrangulation G with \(\chi _f^p(G) < \chi _f(G)\), which we explain in Sect. 1. The other referee suggested the problem in the end of Sect. 4. This work was supported by JSPS KAKENHI, Grant-in-Aid for Scientific Research(C), Grant Number 18K03391.

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Authors and Affiliations

  1. Department of Computer and Information Science, Faculty of Science and Technology, Seikei University, 3-3-1 Kichijoji-Kitamachi, Musashino, Tokyo, 180-8633, Japan

    Kengo Enami

  2. Faculty of Environment and Information Sciences, Yokohama National University, 79-7 Tokiwadai, Hodogaya-ku, Yokohama, 240-8501, Japan

    Kenta Ozeki

  3. College of Engineering Science, Yokohama National University, 79-7 Tokiwadai, Hodogaya-ku, Yokohama, 240-8501, Japan

    Tomoki Yamaguchi

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  1. Kengo Enami
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  2. Kenta Ozeki
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  3. Tomoki Yamaguchi
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Correspondence to Kenta Ozeki.

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This paper is dedicated to Professor Hikoe Enomoto on the occasion of his 75th birthday.

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Enami, K., Ozeki, K. & Yamaguchi, T. Proper Colorings of Plane Quadrangulations Without Rainbow Faces. Graphs and Combinatorics 37, 1873–1890 (2021). https://doi.org/10.1007/s00373-021-02350-5

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  • DOI: https://doi.org/10.1007/s00373-021-02350-5

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