Interval edge-colorings of cubic graphs

TL;DR

This paper investigates interval edge-colorings of cubic graphs, establishing upper bounds on the number of colors needed and demonstrating that these bounds are tight, with special results for bipartite subcubic multigraphs.

Contribution

It proves new upper bounds on the number of colors for interval edge-colorings of cubic graphs and bipartite subcubic multigraphs, including sharpness of these bounds.

Findings

For connected cubic multigraphs, t ≤ |V(G)| + 1.

For connected cubic graphs (not K4), t ≤ |V(G)| - 1.

Bipartite subcubic multigraphs have an interval edge-coloring with at most four colors.

Abstract

An edge-coloring of a multigraph G with colors 1,2,...,t is called an interval t-coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. In this paper we prove that if G is a connected cubic multigraph (a connected cubic graph) that admits an interval t-coloring, then t\leq |V(G)| +1 (t\leq |V(G)|), where V(G) is the set of vertices of G. Moreover, if G is a connected cubic graph, G\neq K_{4}, and G has an interval t-coloring, then t\leq |V(G)| -1. We also show that these upper bounds are sharp. Finally, we prove that if G is a bipartite subcubic multigraph, then G has an interval edge-coloring with no more than four colors.