Interval colorings of complete balanced multipartite graphs

TL;DR

This paper characterizes when complete balanced k-partite graphs can be interval edge-colored and provides bounds on the number of colors needed for such colorings.

Contribution

It establishes necessary and sufficient conditions for interval colorability of complete balanced k-partite graphs and determines the range of t for which interval t-colorings exist.

Findings

Complete balanced k-partite graphs are interval colorable iff nk is even.

Interval t-colorings exist for t in a specific range when nk is even.

For certain k, all t in a range admit an interval t-coloring.

Abstract

A graph $G$ is called a complete $k$-partite ($k\geq 2$) graph if its vertices can be partitioned into $k$ independent sets $V_{1},...,V_{k}$ such that each vertex in $V_{i}$ is adjacent to all the other vertices in $V_{j}$ for $1\leq i<j\leq k$. A complete $k$-partite graph $G$ is a complete balanced $k$-partite graph if $|V_{1}| = |V_{2}| =... = |V_{k}|$. An edge-coloring of a graph $G$ with colors $1,...,t$ 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 $G$ has an interval $t$-coloring for some positive integer $t$. In this paper we show that a complete balanced $k$-partite graph $G$ with $n$ vertices in each part is interval colorable if and only if $nk$ is even. We also prove that if $nk$ is even and $(k-1)n\leq t\leq ((3/2)k-1)n-1$, then a complete balanced $k$-partite graph $G$ admits an interval $t$-coloring. Moreover, if $k=p2^{q}$, where $p$ is odd and $q\in \mathbb{N}$, then a complete balanced $k$-partite graph $G$ has an interval $t$-coloring for each positive integer $t$ satisfying $(k-1)n\leq t\leq (2k-p-q)n-1$.