The interval coloring impropriety of planar graphs

TL;DR

This paper investigates the interval coloring impropriety of planar graphs, proving bounds for outerplanar graphs and showing unboundedness for k-trees, thus resolving two open questions in graph coloring theory.

Contribution

It proves that outerplanar graphs have an interval coloring impropriety of at most 2, confirming a conjecture, and demonstrates that for k-trees, this impropriety is unbounded, refuting another conjecture.

Findings

Outerplanar graphs have   interval coloring impropriety.

Interval coloring impropriety of k-trees is unbounded for k .

Confirmed conjecture for outerplanar graphs, refuted for k-trees.

Abstract

For a graph $G$, we call an edge coloring of $G$ an \textit{improper} \textit{interval edge coloring} if for every $v\in V(G)$ the colors, which are integers, of the edges incident with $v$ form an integral interval. The \textit{interval coloring impropriety} of $G$, denoted by $\mu_{int}(G)$, is the smallest value $k$ such that $G$ has an improper interval edge coloring where at most $k$ edges of $G$ with a common endpoint have the same color. The purpose of this note is to communicate solutions to two previous questions on interval coloring impropriety, mainly regarding planar graphs. First, we prove $\mu_{int}(G) \leq 2$ for every outerplanar graph $G$. This confirms the conjecture by Casselgren and Petrosyan in the affirmative. Secondly, we prove that for each $k\geq 2$, the interval coloring impropriety of $k$-trees is unbounded. This refutes the conjecture by Carr, Cho, Crawford, Ir\v{s}i\v{c}, Pai and Robinson.