On‐line and first fit colorings of graphs

Abstract

A graph coloring algorithm that immediately colors the vertices taken from a list without looking ahead or changing colors already assigned is called "on-line coloring." The properties of on-line colorings are investigated in several classes of graphs. In many cases w e find on-line colorings that use no more colors than some function of the largest clique size of the graph. We show that the first fit on-line coloring has an absolute performance ratio of two for the complement of chordal graphs. We prove an upper bound for the performance ratio of the first fit coloring on interval graphs. It is also shown that there are simple families resisting any on-line algorithm: no on-line algorithm can color all trees by a bounded number of colors.

On some packing problem related to dynamic storage allocation

“…The DPU problem can be also considered in graph-theoretic setting, which has been explored by Gyârfâs and Lehel [8]. Rectangles may be treated as vertices of an interval graph.…”

“…The strongest upper bound [8] (due to W. Just) states that for each interval graph the number of colors used by First Fit is O (n log n), where n is the size of the largest clique.…”

Recognition of split-graphic sequences

“…E.g. Erdős and Gyárfás [18], Gyárfás and Lehel [29] deal with coloring of psplit graphs. Yin and Li [90] gave sufficient conditions for graphic sequences to have a realization with prescribed clique size.…”

An Incremental Model for Combinatorial Maximization Problems

“…Therefore the competitive ratio may be lower than what can reasonably be expected by an on-line algorithm attempting to be fair at each time step. On-line algorithms have been studied in many contexts, including bin packing [5], graph coloring [6], and bipartite matching [7]. Analysis of these on-line algorithms could benefit from theoretical results on the corresponding off-line incremental problem.…”