Interval colorings of graphs -- coordinated and unstable no-wait schedules

TL;DR

This paper investigates interval colorings of graphs, proving that the interval thickness of any graph on n vertices is o(n), and explores the implications for no-wait scheduling in conference coordination, revealing new bounds and unstable configurations.

Contribution

It establishes that the interval thickness of any graph is o(n), improving previous bounds, and constructs bipartite graphs with arbitrarily large gaps in their interval spectrum, linking graph theory to scheduling problems.

Findings

Interval thickness s(G) = o(n) for any graph G on n vertices.

Constructed bipartite graphs with arbitrarily many large gaps in their interval spectrum.

Any conference with n participants can be scheduled in o(n) no-wait periods.

Abstract

A proper edge-coloring of a graph is an interval coloring if the labels on the edges incident to any vertex form an interval of consecutive integers. Interval thickness s(G) of a graph G is the smallest number of interval colorable graphs edge-decomposing G. We prove that s(G)=o(n) for any graph G on n vertices. This improves the previously known bound of 2n/5 by Asratian, Casselgren, and Petrosyan. While we do not have a single example of a graph with interval thickness strictly greater than 2, we construct bipartite graphs whose interval spectrum has arbitrarily many arbitrarily large gaps. Here, an interval spectrum of a graph is the set of all integers t such that the graph has an interval coloring using t colors. Interval colorings of bipartite graphs naturally correspond to no-wait schedules, say for parent-teacher conferences, where a conversation between any teacher and any parent lasts the same amount of time. Our results imply that any such conference with $n$ participants can be coordinated in o(n) no-wait periods. In addition, we show that for any integers t and T, t<T, there is a set of pairs of parents and teachers wanting to talk to each other, such that any no-wait schedules are unstable -- they could last t hours and could last T hours, but there is no possible no-wait schedule lasting x hours if t<x<T.