On the interval coloring impropriety of graphs

An improper interval (edge) coloring of a graph \(G\) is an assignment of integer 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. 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, \(G\odot H\).