3 research outputs found
On Edge Coloring of Multigraphs
Let and be the maximum degree and chromatic index of a
graph , respectively.
Appearing in different format, Gupta\,(1967), Goldberg\,(1973),
Andersen\,(1977), and Seymour\,(1979) made the following conjecture: Every
multigraph satisfies ,
where is the density of . In this
paper, we present a polynomial-time algorithm for coloring any multigraph with
many colors, confirming the conjecture
algorithmically. Since , this
algorithm gives a proper edge coloring that uses at most one more color than
the optimum. As determining the chromatic index of an arbitrary graph is
-hard, the bound is best possible for
efficient proper edge coloring algorithms on general multigraphs, unless