The Local Nature of List Colorings for Graphs of High Girth

  • Authors:

  • Flavio Chierichetti;Andrea Vattani

  • Affiliations:

  • CS Department, Sapienza University of Rome, ;CSE Department, University of California San Diego,

  • Venue:
  • ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
  • Year:
  • 2008

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Abstract

We consider list coloring problems for graphs $\mathcal{G}$ ofgirth larger than c logΔ-1n, where n and Δ ≥ 3 are,respectively, the order and the maximum degree of $\mathcal{G}$,and c is 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 ofBollobás 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 Erdös, we showthat our algorithm can compute k-list vertex colorings ofgraphs having girth larger than clogk-1 n.