[2412.17193]
Israel R. Curbelo
We study the online coloring of $σ$-interval graphs, which are interval graphs with interval lengths in $[1,σ]$ and 2-count interval graphs, which are interval graphs that require at most two distinct interval lengths. For $σ$-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 $σ>1$ such that there is no online algorithm for $σ$-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.