1,646 research outputs found
Improper interval edge colorings of graphs
A -improper edge coloring of a graph is a mapping
such that at most edges of with
a common endpoint have the same color. An improper edge coloring of a graph
is called an improper interval edge coloring if the colors of the edges
incident to each vertex of form an integral interval. In this paper we
introduce and investigate a new notion, the interval coloring impropriety (or
just impropriety) of a graph defined as the smallest such that has
a -improper interval edge coloring; we denote the smallest such by
. We prove upper bounds on for
general graphs and for particular families such as bipartite, complete
multipartite and outerplanar graphs; we also determine
exactly for 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 , there exists a graph with
. 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
- …