On the on-line coloring of unit interval graphs with proper interval representation

TL;DR

The work addresses the online coloring of unit interval graphs under a known proper interval representation by a Builder–Algorithm game, focusing on the quantity $R(\omega)$ that bounds the worst-case number of colors forced by Builder for graphs with clique number at most $\omega$. The authors develop a state-based framework with a separation strategy to iteratively force the game into a final gg state, and they decompose the proof into a sequence of targeted state transitions lemmas and symmetry arguments that culminate in $R(4)=7$. This yields a tight bound for the case $\omega=4$, and, combined with known bounds, tightens the understanding of how representation knowledge impacts online coloring on unit interval graphs. The results contribute to the theory of online graph coloring by clarifying the power of a known proper interval representation in pushing competitive ratios beyond First-Fit benchmarks.

Abstract

We define the problem as a two-player game between Algorithm and Builder. The game is played in rounds. Each round, Builder presents an interval that is neither contained in nor contains any previously presented interval. Algorithm immediately and irrevocably assigns the interval a color that has not been assigned to any interval intersecting it. The set of intervals form an interval representation for a unit interval graph and the colors form a proper coloring of that graph. For every positive integer $ω$, we define the value $R(ω)$ as the maximum number of colors for which Builder has a strategy that forces Algorithm to use $R(ω)$ colors with the restriction that the unit interval graph constructed cannot contain a clique of size $ω+1$. In 1981, Chrobak and Ślusarek showed that $R(ω)\leq2ω-1$. In 2005, Epstein and Levy showed that $R(ω)\geq\lfloor{3ω/2\rfloor}$. This problem remained unsolved for $ω\geq 3$. In 2023, Biró and Curbelo showed that $R(3)=5$. In this paper, we show that $R(4)=7$

Figures (12)

• Figure 1: States
• Figure 2: Outline of the proof of Theorem \ref{['the:main']}.
• Figure 3: $(\mathop{\mathrm{\mathbf{bd}}}\nolimits\rightarrow\mathop{\mathrm{\mathbf{gg}}}\nolimits)$
• Figure 4: $\mathop{\mathrm{\mathbf{ed}}}\nolimits\rightarrow\mathop{\mathrm{\mathbf{gg}}}\nolimits$
• Figure 5: $\mathop{\mathrm{\mathbf{abcab}}}\nolimits\rightarrow\mathop{\mathrm{\mathbf{bd}}}\nolimits$

Theorems & Definitions (5)

• Theorem 1.1
• Proposition 2.1
• Proposition 2.2
• Lemma 2.3
• proof