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

A 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 $θ (G)$ of a graph $G$ is the smallest number of interval colorable graphs edge‐decomposing $G$ . We prove that $θ (G) = o (n)$ for any graph $G$ on n vertices. This improves the previously known bound of $2 [n/5]$, 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 $t$ 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 $t<x<T$