On the performance of the first-fit coloring algorithm on permutation graphs

In this paper we study the performance of a particular on-line coloring algorithm, the First-Fit or Greedy algorithm, on a class of perfect graphs namely the permutation graphs. We prove that the largest number of colors ̄FF(G) that the First-Fit coloring algorithm (FF) needs on permutation graphs of chromatic number ̄(G) = ̄ when taken over all possible vertex orderings is not linearly bounded in terms of the off-line optimum, if ̄ is a fixed positive integer. Specifically, we prove that for any integers ̄ > 0 and k ≥ 0, there exists a permutation graph G on n vertices such that ̄(G) = ̄ and ̄FF(G) ≥ 1 / 2 (( ̄2 + ̄) + k ( ̄2 ̄)), for sufficiently large n. Our result shows that the class of permutation graphs P is not first-fit ̄-bounded; that is, there exists no function f such that for all graphs G ∈ P , ̄FF(G) ≤ f(ˆ(G)). Recall that for perfect graphs ˆ(G) = ̄(G), where ˆ(G) denotes the clique number of G.