Vizing's Theorem states that any graph G has chromatic index either the maximum degree A ( G ) o r A ( G ) + 1 . If G has 2s i -1 points and A(G) = 2s, a well-known necessary condition for the chromatic index to equal 2s is that G have at most 2s' lines. Hilton conjectured that this condition is also sufficient.We present a proof of that conjecture and a corollary that helps determine the chromatic index of some graphs with 2s points and maximum degree 2s -2.

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