Optimal Online Edge Coloring of Planar Graphs with Advice

We study the amount of knowledge about the future that an online algorithm needs to color the edges of a graph optimally (i.e. using as few colors as possible). Assume that along with each edge, the online algorithm receives a fixed number of advice bits. For graphs of maximum degree ∆, it follows from Vizing’s Theorem that ⌈log(∆+ 1)⌉ bits per edge suffice to achieve optimality. We show that even for bipartite graphs, Ω(log∆) bits per edge are in fact necessary. However, we also show that there is an online algorithm which can color the edges of a d-degenerate graph optimally using O(log d) bits of advice per edge. It follows that for planar graphs and other graph classes of bounded degeneracy, only O(1) bits per edge are needed, independently of how large ∆ is.