First-Fit coloring of bounded tolerance graphs

Let G = ( V , E ) be a graph. A tolerance representation of G is a set I = { I v : v ∈ V } of intervals and a set t = { t v : v ∈ V } of nonnegative reals such that x y ∈ E iff I x ∩ I y ≠ 0̸ and ‖ I x ∩ I y ‖ ≥ min { t x , t y } ; in this case G is a tolerance graph. We refine this definition by saying that G is a p -tolerance graph if t v / | I v | ≤ 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 ) ≤ 8 ⌈ 1 1 − p ⌉ ω ( G ) . Also, we show how restricting p forbids induced copies of K s , s . Finally, we observe that there exist 1-tolerance graphs G with ω ( G ) = 2 and arbitrarily large Grundy number.