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

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 χ_{FF}(G). The exact value of R := \sup_G [χ_{FF}(G) / ω(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 Ślusarek (1988) showed R >= 4, and Ślusarek (1993) improved this to 4.45. We prove R >= 5.

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