On-line coloring of Is-free graphs and co-planar graphs

An on-line vertex coloring algorithm receives vertices of a graph in some externally determined order. Each new vertex is presented together with a set of the edges connecting it to the previously presented vertices. As a vertex is presented, the algorithm assigns it a color which cannot be changed afterwards. The on-line coloring problem was addressed for many different classes of graphs defined in terms of forbidden structures. We analyze the class of Isfree graphs, i.e., graphs in which the maximal size of an independent set is at most s − 1. An old Szemerédi’s result implies that for each on-line algorithm A there exists an on-line presentation of an Is-free graph G forcing A to use at least s 2 χ(G) colors. We prove that any greedy algorithm uses at most s 2 χ(G) colors for any on-line presentation of any Is-free graph G. Since the class of co-planar graphs is a subclass of I5-free graphs all greedy algorithms use at most 5 2 χ(G) colors for co-planar G’s. We prove that, even in a smaller class, this is an almost tight bound.