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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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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DOI: https://doi.org/10.1007/s00373-014-1438-9