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13.
Sufficient sparseness conditions for ▫$G^2$▫ to be ▫$(\Delta + 1)$▫-choosable, when ▫$\Delta \ge 5$▫
Cranston, Daniel W. ; Škrekovski, Riste
We determine the list chromatic number of the square of a graph ▫$\chi_\ell(G^2)$▫ in terms of its maximum degree ▫$\Delta$▫ when its maximum average degree, denoted ▫$\text{mad}(G)$▫, is ...sufficiently small. For ▫$\Delta \ge 6$▫, if ▫$\text{mad}(G) < 2 + \frac{4\Delta - 8}{5\Delta + 2}$▫, then ▫$\chi_\ell(G^2) = \Delta + 1$▫. In particular, if ▫$G$▫ is planar with girth ▫$g \ge 7 + \frac{12}{\Delta - 2}$▫, then ▫$\chi_\ell(G^2) = \Delta + 1$▫. Under the same conditions, ▫$\chi_\ell^i(G^2) = \Delta$▫, where ▫$\chi_\ell^i$▫ is the list injective chromatic number.
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