Very fast parallel algorithms for approximate edge coloring Original Research Article

This paper presents very fast parallel algorithms for approximate edge coloring. Let log(1) n=log n, log(k) n=log(log(k−1) n), and log∗(n)=min{k | log(k) n<1}. It is shown that a graph with n vertices and m edges can be edge colored with (2⌈log1/4 log∗(n)⌉)c·(⌈Δ/logc/4 log∗(n)⌉)2 colors in O(log log∗(n)) time using O(m+n) processors on the EREW PRAM, where Δ is the maximum vertex degree of the graph and c is an arbitrarily large constant. It is also shown that the graph can be edge colored using at most ⌈4Δ1+4/log log log∗(Δ) log1/2 log∗ (Δ)⌉ colors in O(log Δ log log∗(Δ)/log log log∗ (Δ)+log log∗(n)) time using O(m+n) processors on the same model.