On Improperly Chromatic-Choosable Graphs (original) (raw)

Abstract

A graph is \(d\)-improperly chromatic-choosable if its \(d\)-improper choice number coincides with its \(d\)-improper chromatic number. For fixed \(d\ge 0\), we show that if the \(d\)-improper chromatic number is close enough to \(\frac{1}{d+1}\) of the number of vertices in \(G\), then \(G\) is \(d\)-improperly chromatic-choosable. As a consequence, we show that the join \(G + K_n\) is \(d\)-improperly chromatic-choosable when \(n\ge (|V(G)|+d)^2\). We also raise a conjecture on \(d\)-improper chromatic-choosability.

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Acknowledgments

We thank the referees for their constructive comments. In particular, the current form of Conjecture 1 as a natural generalization of Ohba’s conjecture is inspired by one of the referees.

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Authors and Affiliations

  1. College of Information Engineering, Tarim University, Alar, 843300, China
    Zhidan Yan, Wei Wang & Nini Xue

Authors

  1. Zhidan Yan
  2. Wei Wang
  3. Nini Xue

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Correspondence toZhidan Yan.

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Yan, Z., Wang, W. & Xue, N. On Improperly Chromatic-Choosable Graphs.Graphs and Combinatorics 31, 1807–1814 (2015). https://doi.org/10.1007/s00373-014-1438-9

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