First-fit coloring on interval graphs has performance ratio at least 5

Abstract

First-fit is the online graph coloring algorithm that considers vertices one at a time in some order and assigns each vertex the least positive integer not used already on a neighbor. The maximum number of colors used by first-fit on graph G over all vertex orders is denoted \chi_{FF}(G). The exact value of R := \sup_G [\chi_{FF}(G) / \omega(G)] over interval graphs G is unknown. Pemmaraju, Raman, and Varadarajan (2004) proved R <= 10, and this can be improved to 8. Witsenhausen (1976) and Chrobak and \'Slusarek (1988) showed R >= 4, and \'Slusarek (1993) improved this to 4.45. We prove R >= 5.