The Tournament Scheduling Problem with Absences

TL;DR

This paper introduces a new graph coloring problem modeling tournament scheduling with absences, analyzing worst-case rounds needed based on graph structure and absence information, with conjectures and bounds for various graph classes.

Contribution

It formulates the tournament scheduling with absences as a novel graph coloring problem, proposes conjectures relating to upper bounds, and provides partial proofs and bounds for different graph types.

Findings

Conjecture that hi^t(G)=elta(G)+2t for all graphs.

Conjecture that hi_{OL}^t(G)=hi'(G)+2t for online scheduling.

Established bounds and partial proofs for bipartite and complete graphs.

Abstract

We study time scheduling problems with allowed absences as a new kind of graph coloring problem. One may think of a sport tournament where each player (each team) is permitted a certain number $t$ of absences. We then examine how many rounds are needed to schedule the whole tournament in the worst case. This upper limit depends on $t$ and on the structure of the graph $G$ whose edges represent the games that have to be played, but also on whether or not the absences are announced before the tournament starts. Therefore, we actually have two upper limits for the number of required rounds. We have $\chi^t(G)$ for pre-scheduling if all absences are pre-fixed, and we have $\chi_{\textit{OL}}^t(G)$ for on-line scheduling if we have to stay flexible and deal with absences when they occur. We conjecture that $\chi^t(G)=\Delta(G)+2t$ and that $\chi_{\textit{OL}}^t(G)=\chi'(G)+2t.$ The first conjecture is stronger than the Total Coloring Conjecture while the second is weaker than the On-Line List Edge Coloring Conjecture. Our conjectures hold for all bipartite graphs. For complete graphs, we prove them partially. Lower and upper bounds to $\chi^t(G)$ and $\chi_{\textit{OL}}^t(G)$ for general multigraphs $G$ are established, too.