Asymptotics of the total chromatic number for multigraphs

For loopless multigraphs, the total chromatic number is asymptotically its fractional counterpart as the latter invariant tends to infinity. The proof of this is based on a recent theorem of Kahn establishing the analogous asymptotic behaviour of the list-chromatic index for multigraphs. The total colouring conjecture, proposed independently by Behzad [1] and Vizing [11], asserts that the total chromatic number X t of a simple graph exceeds the maximum degree ~ by at most two. The most recent increment (better: giant leap) toward a proof of this conjecture was made by Molloy and Reed [8], who established by probabilistic means that the difference between X t and ~ is at most a constant ( say c). An immediate consequence of their result is that for simple graphs, X t is asymptotically its fractional analogue X: as the latter tends to infinity: for this follows from ~ + 1 :::; X: :::; X t :::; ~ + c. This leads naturally to the following question: does X t enjoy the same asymptotic connection with X: for loopless multigraphs (henceforth multigraphs)? That this question has an affirmative answer was conjectured in [6]. The purpose of this note is to verify that conjecture: Theorem 1 For multigraphs,