Decomposing graphs into interval colorable subgraphs and no-wait multi-stage schedules

TL;DR

This paper introduces the concept of interval coloring thickness, a measure of how a graph can be decomposed into interval colorable subgraphs, with applications in scheduling and timetabling.

Contribution

It defines the new notion of interval coloring thickness and provides bounds and conditions for graphs to be decomposed into interval colorable subgraphs.

Findings

Connected 3-edge colorable graphs with max degree 3 are interval colorable.

Upper bounds on interval coloring thickness for general graphs.

Improved bounds for bipartite, planar, and complete multipartite graphs.

Abstract

A graph $G$ is called interval colorable if it has a proper edge coloring with colors $1,2,3,\dots$ such that the colors of the edges incident to every vertex of $G$ form an interval of integers. Not all graphs are interval colorable; in fact, quite few families have been proved to admit interval colorings. In this paper we introduce and investigate a new notion, the interval coloring thickness of a graph $G$, denoted ${\theta_{\mathrm{int}}}(G)$, which is the minimum number of interval colorable edge-disjoint subgraphs of $G$ whose union is $G$. Our investigation is motivated by scheduling problems with compactness requirements, in particular, problems whose solution may consist of several schedules, but where each schedule must not contain any waiting periods or idle times for all involved parties. We first prove that every connected properly $3$-edge colorable graph with maximum degree $3$ is interval colorable, and using this result, we deduce an upper bound on ${\theta_{\mathrm{int}}}(G)$ for general graphs $G$. We demonstrate that this upper bound can be improved in the case when $G$ is bipartite, planar or complete multipartite and consider some applications in timetabling.