Abstract
A k-improper edge coloring of a graph G is a mapping α:E(G)⟶N such that at most k edges of G with a common endpoint have the same color. An improper edge coloring of a graph G is called an improper interval edge coloring if the colors of the edges incident to each vertex of G form an integral interval. In this paper we introduce and investigate a new notion, the interval coloring impropriety (or just impropriety) of a graph G defined as the smallest k such that G has a k-improper interval edge coloring; we denote the smallest such k by μint(G). We prove upper bounds on μint(G) for general graphs G and for particular families such as bipartite, complete multipartite and outerplanar graphs; we also determine μint(G) exactly for G belonging to some particular classes of graphs. Furthermore, we provide several families of graphs with large impropriety; in particular, we prove that for each positive integer k, there exists a graph G with μint(G)=k. Finally, for graphs with at least two vertices we prove a new upper bound on the number of colors used in an improper interval edge coloring.
Highlights
A proper edge coloring of a graph G is called an interval t-coloring if exactly t colors appear on the edges of G and the colors of the edges incident to every vertex v of G form an interval of integers
We consider the number of colors in an improper interval edge coloring and obtain a new upper bound on the number of colors used in such a coloring
In [8], the authors considered the problem of constructing interval edge colorings of so-called generalized θ -graphs; a generalized θ -graph, denoted by θm, is a graph consisting of two vertices u and v together with m internally-disjoint (u, v)-paths, where 2 ≤ m < ∞
Summary
A proper edge coloring of a graph G is called an interval t-coloring if exactly t colors appear on the edges of G and the colors of the edges incident to every vertex v of G form an interval of integers This notion was introduced by Asratian and Kamalian [3] (available in English as [4]), motivated by the problem of constructing timetables without ‘‘gaps’’ for teachers and classes. Called an improper interval (edge) coloring if the colors on the edges incident to every vertex of the graph form a set of consecutive integers This edge coloring model seems to have been first considered by Hudak et al [18], their investigation has a different focus than ours. We consider the number of colors in an improper interval edge coloring and obtain a new upper bound on the number of colors used in such a coloring
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