On edge-colorings of graphs.

Improved bounds for the chromatic index of graphs and multigraphs

“…The following lemma is a localized version of a slight generalization of the bound of Andersen and Goldberg [1,8], and its proof is an adaptation of Goldberg's proof in [8]. As we will see, this lemma has interesting implications.…”

“…It is easy to see that (G) ≤ max{∆(G), ζ(G)} and, consequently, Theorem 1 implies the Andersen-Goldberg bound [1,8], which in turn implies the Shannon bound [16] and Vizing bound [19] for χ (G). A careful examination reveals that Theorem 1 also implies a result of Berge and Fournier [2].…”

“…We now present several standard results on edge-coloring [1,8], which are used in the following sections.…”

“…More recently, Andersen [1] and Goldberg [8] independently showed that χ (G) ≤ max{∆(G), ζ(G)}, where…”

Properly colored cycles in edge-colored graphs

“…One class consists of research in which the purpose is to assign the colors to the edges of a graph with certain requirements or restrictions. A typical example is determining the edge-chromatic number [6], i.e., deciding the minimum number of colors that is needed to color the edges of a graph such that each pair of adjacent edges are assigned different colors. The other class consists of research in which the goal is to show the existence of (or find) subgraphs with certain coloring characteristics in a graph that has already been edge-colored.…”