On the existence of a matching orthogonal to a 2-factorization

Let us recall the following problem which was posed by Brian Alspach [l]. Let FI, F2, . . . , Fd be any 2-factorization of a 2d regular graph G. Is it possible to find a d-matching M in G such that M contains precisely one edge from each of F,, 4, . . . , F,? Clearly G is of order n 2 2d + 1 and an easy counting argument shows that the answer is yes if n 2 4d 3. This bound was slightly improved to 4d 5 by G. Liu (see [2]). In this note we answer positively the question for graphs of order n > 3.23d. A particular case of this problem was also formulated by F. Chung (oral communication), who conjectured that any graph G which is the edgedisjoint union of d hamiltonian cycles contains a d-matching orthogonal to these cycles, i.e. containing precisely one edge from each cycle. Let us introduce some notation and definitions that will be used later. For any set X, 1x1 will denote its cardinality. Let G be a 2d-regular graph. Let 4, & l l . , Fd be any 2-factorization of G. If X and Y are two subsets of V(G), then, for any integers iI, i2, . . . , ij, between 1 and d, we will denote by Et,,12,..., i (X, Y) the set of edges of 6, U E, U l l l U ei with one end in X and the other end in Y.