On the chromatic index of multigraphs and a conjecture of Seymour (I)

Abstract

AbstractLet g = (V, E, w) be a multigraph, where V is a set of vertices, E is a set of edges, and w is a vector of edge multiplicities. It is well known that ϱ, the maximum degree of g, is a lower bound on the cardinality of a proper edge coloring of g. Another lower bound is given by κ = max{w(E(S))((|S| − 1)2) • S ⊆ V, |S| odd and |S| ≠ 1}, where w(E(S)) is the number of edges both ends of which belong to S. P. D. Seymour [Proc. London Math. Soc. (3) 38 (1979), 423–460] has made the conjecture that the minimum number of colors in a proper edge coloring of g is less than or equal to max {ϱ + 1, ⌜κ⌝}, where ⌜κ⌝ denotes the least integer greater than or equal to κ. In this paper we show that Seymour's conjecture can be reduced to a conjecture about critical nonseparable graphs (in the sense of matching theory). We also show that the latter conjecture is verified in the case of outerplanar graphs, thus proving that Seymour's conjecture holds for outerplanar graphs