On the interval coloring impropriety of graphs

TL;DR

This paper investigates the interval coloring impropriety of various graph classes, providing constructions, bounds, and supporting evidence for a conjecture that this impropriety is at most 2 for complete multipartite graphs.

Contribution

It constructs interval colorings for certain graphs, improves bounds on impropriety for specific classes, and supports a conjecture regarding complete multipartite graphs.

Findings

Constructed interval colorings for a subclass of complete multipartite graphs.

Provided improved upper bounds for 2-trees, iterated triangulations, and outerplanar graphs.

Supported the conjecture that the impropriety is at most 2 for all complete multipartite graphs.

Abstract

An improper interval (edge) coloring of a graph $G$ is an assignment of colors to the edges of $G$ satisfying the condition that, for every vertex $v \in V(G)$, the set of colors assigned to the edges incident with $v$ forms an integral interval. An interval coloring is $k$-improper if at most $k$ edges with the same color all share a common endpoint. The minimum integer $k$ such that there exists a $k$-improper interval coloring of the graph $G$ is the interval coloring impropriety of $G$, denoted by $\mu_{int}(G)$. In this paper, we provide a construction of an interval coloring of a subclass of complete multipartite graphs. This provides additional evidence to the conjecture by Casselgren and Petrosyan that $\mu_{int}(G)\leq 2$ for all complete multipartite graphs $G$. Additionally, we determine improved upper bounds on the interval coloring impropriety of several classes of graphs, namely 2-trees, iterated triangulations, and outerplanar graphs. Finally, we investigate the interval coloring impropriety of the corona product of two graphs, $G\odot H$.