Abstract
Proportional choosability is a list analogue of equitable coloring that was introduced in 2019. The smallest k for which a graph G is proportionally k-choosable is the proportional choice number of G, and it is denoted \(\chi _{pc}(G)\). In the first ever paper on proportional choosability, it was shown that when \(2 \le n \le m\), \(\max \{ n + 1, 1 + \lceil m / 2 \rceil \} \le \chi _{pc}(K_{n,m}) \le n + m - 1\). In this note we improve on this result by showing that \(\max \{ n + 1, \lceil n / 2 \rceil + \lceil m / 2 \rceil \} \le \chi _{pc}(K_{n,m}) \le n + m -1- \lfloor m/3 \rfloor\). In the process, we prove some new lower bounds on the proportional choice number of complete multipartite graphs. We also present several interesting open questions.
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Notes
We work under the assumption that a proper k-coloring has exactly k, possibly empty, color classes.
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Acknowledgements
The authors would like to thank Hemanshu Kaul for many helpful conversations. The authors would also like to thank the anonymous referee for helpful comments on this paper
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Mudrock, J.A., Hewitt, J., Shin, P. et al. Proportional Choosability of Complete Bipartite Graphs. Graphs and Combinatorics 37, 381–392 (2021). https://doi.org/10.1007/s00373-020-02255-9
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DOI: https://doi.org/10.1007/s00373-020-02255-9