Abstract
Vertex coloring is a central concept in graph theory and an important symmetry-breaking primitive in distributed computing. Whereas degree-Δ graphs may require palettes of Δ+1 colors in the worst case, it is well known that the chromatic number of many natural graph classes can be much smaller. In this paper we give new distributed algorithms to find (Δ/k)-coloring in graphs of girth 4 (triangle-free graphs), girth 5, and trees. The parameter k can be at most (14−o(1))lnΔ in triangle-free graphs and at most (1−o(1))lnΔ in girth-5 graphs and trees, where o(1) is a function of Δ. Specifically, for Δ sufficiently large we can find such a coloring in O(k+log⁎n) time. Moreover, for any Δ we can compute such colorings in roughly logarithmic time for triangle-free and girth-5 graphs, and in O(logΔ+logΔlogn) time on trees. As a byproduct, our algorithm shows that the chromatic number of triangle-free graphs is at most (4+o(1))ΔlnΔ, which improves on Jamall's recent bound of (67+o(1))ΔlnΔ. Finally, we show that (Δ+1)-coloring for triangle-free graphs can be obtained in sublogarithmic time for any Δ.
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