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 is an element of V(G) 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)(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) <= 2 for every outerplanar graph G. This confirms a conjecture by Casselgren and Petrosyan in the affirmative. Secondly, we prove that for each k >= 2, the interval coloring impropriety of k-trees is unbounded. This refutes a conjecture by Carr, Cho, Crawford, Ir & scaron;i & ccaron;, Pai and Robinson. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.