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.
Categories: math.CO