The interval coloring impropriety of planar graphs

For a graph $G$, we call an edge coloring of $G$ an improper interval edge coloring if for every $v\in V\left(G\right)$ the colors, which are integers, of the edges incident with $v$ form an integral interval. The interval coloring impropriety of $G$, denoted by $\mu int\left(G\right)$, 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 $\mu int\left(G\right)\le 2$ for every outerplanar graph $G$. This confirms a conjecture by Casselgren and Petrosyan in the affirmative. Secondly, we prove that for each $k\ge 2$, the interval coloring impropriety of $k$-trees is unbounded. This refutes a conjecture by Carr, Cho, Crawford, Iršič, Pai and Robinson.