Some applications of matching theorems

Abstract

This thesis contains the results of two investigations. The rst concerns the 1- factorizability of regular graphs of high degree. Chetwynd and Hilton proved in 1989 that all regular graphs of order 2n and degree 2n where > 1 2 ( p 7 ð€€€ 1) 0:82288 are 1-factorizable. We show that all regular graphs of order 2n and degree 2n where is greater than the second largest root of 4x6 ð€€€ 28x5 ð€€€ 71x4 + 54x3 + 88x2 ð€€€ 62x + 3 ( 0:81112) are 1-factorizable. It is hoped that in the future our techniques will yield further improvements to this bound. In addition our study of barriers in graphs of high minimum degree may have independent applications. The second investigation concerns partial latin squares that satisfy Hall's Condition. The problem of completing a partial latin square can be viewed as a listcolouring problem in a natural way. Hall's Condition is a necessary condition for such a problem to have a solution. We show that for certain classes of partial latin square, Hall's Condition is both necessary and su cient, generalizing theorems of Hilton and Johnson, and Bobga and Johnson. It is well-known that the problem of deciding whether a partial latin square is completable is NP-complete. We show that the problem of deciding whether a partial latin square that is promised to satisfy Hall's Condition is completable is NP-hard.