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 \(v ∈ V(G)\), 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 \(μᵢₙₜ(G)\). 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, \(G H\).