The Local Nature of List Colorings for Graphs of High Girth

We consider list coloring problems for graphs $\mathcal{G}$ ofgirth larger than clogΔ-1n, where nand Δ≥ 3 are,respectively, the order and the maximum degree of $\mathcal{G}$,and cis a suitable constant. First, we determine that theedge and total list chromatic numbers of these graphs are$\chi'_l({\mathcal{G}}) = \Delta$ and $\chi''_l({\mathcal{G}}) =\Delta + 1$. This proves that the general conjectures ofBollobas and Harris (1985), Behzad and Vizing (1969) andJuvan, Mohar and `krekovski (1998) hold for this particular classof graphs. Moreover, our proofs exhibit a certain degree of "locality",which we exploit to obtain an efficient distributed algorithm ableto compute both kinds of optimal list colorings. Also, using an argument similar to one of Erdos, we showthat our algorithm can compute k-list vertex colorings ofgraphs having girth larger than clogk-1n.

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