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Graph edge coloring has a rich theory, many applications and beautiful conjectures, and it is studied not only by mathematicians, but also by computer scientists. In this survey, written for the non-expert, we shall describe some main results and techniques and state some of the many popular conjectures in the theory. Besides known results a new basic result about brooms is obtained. [ABSTRACT FROM AUTHOR]

Abstract: A graph G with maximum degree and edge chromatic number is edge--critical if for every edge e of G. New lower bounds are given for the average degree of an edge--critical graph, which improve on the best bounds previously known for most values of . Examples of edge--critical graphs are also given. In almost all cases, there remains a large gap between the best lower bound known and the smallest average degree of any known edge--critical graph. [Copyright &y& Elsevier]

Let G be a simple graph with average degree d¯ and maximum degree Δ. It is proved, in this paper that G is not critical if d¯⩽6 and Δ⩾8, or d¯⩽20/3 and Δ⩾9. This result generalizes earlier results of Vizing (Metody Diskret. Analiz. 5 (1965) 9), Mel'nikov (Mat. Zametki 7 (1970) 671) and Hind and Zhao (Discrete Math. 190 (1998) 107) and Yan and Zhao (Graphs Combin. 16 (2) (2000) 245). It also improves a result by Fiorini (Math. Proc. Cambridge Philos. Soc. 77 (1975) 475) on the number of edges of critical graphs for certain Δ. [Copyright &y& Elsevier]

Abstract: The known proofs of the famous theorem of Vizing on edge coloring of (multi)graphs are not long but sophisticated. The main goal of this note is to present an auxiliary (multi)digraph that simplifies and facilitates proofs of it. The secondary goal is to use the approach for proofs of Vizing’s Adjacency Lemma and the Andersen–Goldberg Theorem. [Copyright &y& Elsevier]