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

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 no...

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Bibliographic Details
Published in:Discrete Mathematics and Theoretical Computer Science Vol. 27:2; no. Combinatorics; pp. 1 - 16
Main Authors: Curbelo, Israel R., Malko, Hannah R.
Format: Journal Article
Language:English
Published: Nancy DMTCS 01.08.2025
Discrete Mathematics & Theoretical Computer Science
Subjects:
05c15 (primary) 68w27 (secondary)
Algorithms
Coloring
computer science - data structures and algorithms
Graph theory
Mathematical research
mathematics - combinatorics
Representations
United States
ISSN:1365-8050, 1462-7264, 1365-8050
Online Access:Get full text
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Summary: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 $\omega$, we define the value $R(\omega)$ as the maximum number of colors for which Builder has a strategy that forces Algorithm to use $R(\omega)$ colors with the restriction that the unit interval graph constructed cannot contain a clique of size $\omega+1$. In 1981, Chrobak and \'{S}lusarek showed that $R(\omega)\leq2\omega -1$. In 2005, Epstein and Levy showed that $R(\omega)\geq\lfloor{3\omega/2\rfloor}$. This problem remained unsolved for $\omega\geq 3$. In 2023, Bir\'o and Curbelo showed that $R(3)=5$. In this paper, we show that $R(4)=7$ Comment: To be published in Discrete Mathematics & Theoretical Computer Science (DMTCS)
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ISSN:1365-8050
1462-7264
1365-8050
DOI:10.46298/dmtcs.14088