An application of Tutte’s Theorem to 1-factorization of regular graphs of high degree

A well known conjecture in graph theory states that every regular graph of even order 2 n and degree λ ( 2 n ) , where λ ≥ 1 / 2 , is 1-factorizable. Chetwynd and Hilton [A.G. Chetwynd, A.J.W. Hilton, 1-factorizing regular graphs of high degree — An improved bound, Discrete Math. 75 (1989) 103–112] and, independently, Niessen and Volkmann [T. Niessen, L. Volkmann, Class 1 conditions depending on the minimum degree and the number of vertices of maximum degree, J. Graph Theory (2) 14 (1990) 225–246] proved the above conjecture under the assumption that λ ≥ 7 − 1 2 ≈ 5 / 6 . Since these results were published no significant or even partial improvement has been made in terms of lowering the bound on λ . We shall obtain here a partial improvement on λ . Specifically, using the original Chetwynd–Hilton approach and Tutte’s 1-Factor Theorem, we show that the above bound can be improved to λ > 57 − 3 6 ≈ 3 / 4 , apart (possibly) from two families of exceptional cases. We then show, under the stronger assumption that λ ≥ λ ∗ ≈ 0.785 , that one of the two families of exceptional cases cannot occur.