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

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.