Interval non-edge-colorable bipartite graphs and multigraphs

TL;DR

This paper develops methods to construct bipartite graphs and multigraphs that cannot be colored with interval edge-colorings, expanding understanding of the smallest such graphs and addressing open questions in graph coloring theory.

Contribution

The paper introduces new construction methods for bipartite graphs and multigraphs that are not interval edge-colorable, providing explicit examples with specific sizes and degrees.

Findings

Constructed bipartite graphs with no interval coloring of sizes 20, 19, 21 and maximum degrees 11, 12, 13.

Extended the class of known non-interval-colorable bipartite graphs.

Partially answered an open question about the minimal size of such graphs.

Abstract

An edge-coloring of a graph $G$ with colors $1,...,t$ 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 1991 Erd\H{o}s constructed a bipartite graph with 27 vertices and maximum degree 13 which has no interval coloring. Erd\H{o}s's counterexample is the smallest (in a sense of maximum degree) known bipartite graph which is not interval colorable. On the other hand, in 1992 Hansen showed that all bipartite graphs with maximum degree at most 3 have an interval coloring. In this paper we give some methods for constructing of interval non-edge-colorable bipartite graphs. In particular, by these methods, we construct three bipartite graphs which have no interval coloring, contain 20,19,21 vertices and have maximum degree 11,12,13, respectively. This partially answers a question that arose in [T.R. Jensen, B. Toft, Graph coloring problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995, p. 204]. We also consider similar problems for bipartite multigraphs.