Interval colorings of graphs—Coordinated and unstable no‐wait schedules

Abstract

AbstractA proper edge‐coloring of a graph is an interval coloring if the labels on the edges incident to any vertex form an interval, that is, form a set of consecutive integers. The interval coloring thickness of a graph is the smallest number of interval colorable graphs edge‐decomposing . We prove that for any graph on vertices. This improves the previously known bound of , see Asratian, Casselgren, and Petrosyan. While we do not have a single example of a graph with an interval coloring thickness strictly greater than 2, we construct bipartite graphs whose interval coloring spectrum has arbitrarily many arbitrarily large gaps. Here, an interval coloring spectrum of a graph is the set of all integers such that the graph has an interval coloring using 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 participants can be coordinated in  no‐wait periods. In addition, we show that for any integers and , , 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 hours and could last hours, but there is no possible no‐wait schedule lasting hours if .