We consider the on-line coloring problem restricted to proper interval graphs with known interval representation. Chrobak and \'{S}lusarek (1981) showed that the greedy $\textrm{First-Fit}$ algorithm has a strict competitive ratio of $2$. It remains open whether there is an on-line algorithm that performs better than $\textrm{First-Fit}$. Piotr (2008) showed that if the representation is not known, there is no better on-line algorithm. Epstein and Levy (2005) showed that no on-line algorithm has a strict competitive ratio less than $1.5$ when a unit-interval representation is known, which was later improved to $1.\overline{3}$. In this paper, we show that there is no on-line algorithm with strict competitive ratio less than $1.75$ by presenting a strategy that can force any on-line algorithm to use $7$ colors on a proper interval graph $G$ with chromatic number $\chi(G)\leq 4$ and known interval representation.

翻译：我们研究了限制在已知区间表示的真区间图上的在线染色问题。Chrobak和Ślusarek（1981）指出，贪心的“先入先出”（First-Fit）算法具有严格竞争比为2。是否存在比First-Fit性能更优的在线算法仍未解决。Piotr（2008）证明，若区间表示未知，则不存在更优的在线算法。Epstein与Levy（2005）表明，当已知单位区间表示时，任何在线算法的严格竞争比均不小于1.5，该下界后来被改进为1.\overline{3}。本文通过提出一种策略，迫使任意在线算法对色数\(\chi(G)\leq 4\)且已知区间表示的真区间图\(G\)使用7种颜色，从而证明不存在严格竞争比小于1.75的在线算法。