First-Fit coloring of bounded tolerance graphs

Let G=(V,E) be a graph. A tolerance representation of G is a set I=Iv:v∈V of intervals and a set t=tv:v∈V of nonnegative reals such that xy∈E iff Ix∩Iy≠Combining long solidus overlay and ||Ix∩Iy||<mintx,ty; in this case G is a tolerance graph. We refine this definition by saying that G is a p-tolerance graph if tv|Iv|≤p for all v∈V. A Grundy coloring g of G is a proper coloring of V with positive integers such that for every positive integer i, if i<g(v) then v has a neighbor u with g(u)=i. The Grundy number Γ(G) of G is the maximum integer k such that G has a Grundy coloring using k colors. It is also called the First-Fit chromatic number. For fixed 0≤p<1 we prove that if G is a p-tolerance graph then, Γ(G)=Θ(ω(G)1-p), and in particular, Γ(G)≤811-p⌉ω(G). Also, we show how restricting p forbids induced copies of Ks,s. Finally, we observe that there exist 1-tolerance graphs G with ω(G)=2 and arbitrarily large Grundy number.