Edge‐Colorings of Graphs: A Progress Report

It is almost ten years since the appearance of S. Fiorini and R. J. Wilson’s monograph Edge-Colourings of Graphs [21]. In this expository paper we outline some of the progress on edge-colorings since its publication, with particular reference to a recent series of papers by A. G. Chetwynd and A. J. W. Hilton. Throughout the present paper, G will denote a simple connected graph with n vertices, rn edges, and maximum vertex-degree A. The chromatic index x ’ ( G ) of G is the smallest number of colors needed to color the edges of G so that adjacent edges are colored differently. For example, if C, is the cycle graph of order n, then x’(C,) = 2 or 3, according as n is even or odd. The most important result on edge-colorings is due to V. G. Vizing [35], and gives very sharp bounds for x’(C):

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