On the deficiency of complete multipartite graphs

TL;DR

This paper studies the deficiency of complete multipartite graphs, providing tight bounds and exact values for certain classes, to understand how many pendant edges are needed for an interval coloring.

Contribution

It introduces bounds and exact values for the deficiency of complete multipartite graphs, advancing understanding of their interval colorability.

Findings

Established a tight upper bound for the deficiency of complete multipartite graphs.

Determined or bounded the deficiency for specific classes of these graphs.

Enhanced understanding of the relationship between graph structure and interval colorability.

Abstract

An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an \emph{interval $t$-coloring} if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an integer interval. It is well-known that there are graphs that do not have interval colorings. The \emph{deficiency} of a graph $G$, denoted by $\mathrm{def}(G)$, is the minimum number of pendant edges whose attachment to $G$ leads to a graph admitting an interval coloring. In this paper we investigate the problem of determining or bounding of the deficiency of complete multipartite graphs. In particular, we obtain a tight upper bound for the deficiency of complete multipartite graphs. We also determine or bound the deficiency for some classes of complete multipartite graphs.