On interval edge-colorings of bipartite graphs of small order

TL;DR

This paper investigates interval edge-colorings of bipartite graphs, demonstrating that all bipartite graphs with up to 15 vertices are interval colorable, extending previous results and exploring small graph classes.

Contribution

It proves that all bipartite graphs with 15 vertices are interval colorable, expanding the known range beyond graphs with up to 14 vertices.

Findings

All bipartite graphs on 15 vertices are interval colorable.

Interval non-colorable bipartite graphs have at least 19 vertices.

Deciding interval colorability is NP-complete.

Abstract

An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if it has an interval $t$-coloring for some positive integer $t$. The problem of deciding whether a bipartite graph is interval colorable is NP-complete. The smallest known examples of interval non-colorable bipartite graphs have $19$ vertices. On the other hand it is known that the bipartite graphs on at most $14$ vertices are interval colorable. In this work we observe that several classes of bipartite graphs of small order have an interval coloring. In particular, we show that all bipartite graphs on $15$ vertices are interval colorable.