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arXiv · 2412.17193

Online coloring of short interval graphs and two-count interval graphs

Israel R. Curbelo
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Abstract

We study the online coloring of $\sigma$-interval graphs, which are interval graphs with interval lengths in $[1,\sigma]$ and 2-count interval graphs, which are interval graphs that require at most two distinct interval lengths. For $\sigma$-interval graphs, the Kierstead-Trotter algorithm has competitive ratio 3 and no online algorithm has competitive ratio better than 2. In this paper, we show that for every $\varepsilon>0$, there is a $\sigma>1$ such that there is no online algorithm for $\sigma$-interval coloring with competitive ratio less than $3-\varepsilon$. For 2-count interval graphs, we show that the greedy algorithm First-Fit has competitive ratio at most $4$, that there is no online algorithm with competitive ratio less than $2.5$ when the interval representation is unknown, and that there is no online algorithm with competitive ratio less than $2$ when the interval representation is known.

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Israel R. Curbelo. 2024-12-22. Online coloring of short interval graphs and two-count interval graphs. https://arxiv.org/abs/2412.17193

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