Let G be a simple graph with 3#(G) > |V |.TheOverfull Graph Conjecture states that the chromatic index of G is equal to #(G), if G does not contain an induced overfull subgraph H with #(H)=#(G), and otherwise it is equal to #(G) + 1. We present an algorithm that determines these subgraphs in O(n 5/3 m) time, in general, and in O(n 3 ) time, if G is regular. Moreover, it is shown that G can have at most three of these subgraphs. If 2#(G) #|V |,thenG contains at most one of these subgraphs, and our former algorithm for this situation is improved to run in linear time. 1

A chromatic-index-critical graph G on n vertices is non-trivial if it has at most \Deltab n 2 c edges. We prove that there is no chromatic-index-critical graph of order 12, and that there are precisely two non-trivial chromatic index critical graphs on 11 vertices. Together with known results this implies that there are precisely three non-trivial chromaticindex -critical graphs of order 12. 1 Introduction A famous theorem of Vizing [20] states that the chromatic index Ø 0 (G) of a simple graph G is \Delta(G) or \Delta(G) + 1, where \Delta(G) denotes the maximum vertex degree in G. A graph G is class 1 if Ø 0 (G) = \Delta(G) and it is class 2 otherwise. A class 2 graph G is (chromatic index) critical if Ø 0 (G \Gamma e) ! Ø 0 (G) for each edge e of G. If we want to stress the maximum vertex degree of a critical graph G we say G is \Delta(G)-critical. Critical graphs of odd order are easy to construct while not much is known about critical graphs of even order. One reas...