Remarks on regular factors in vertex-deleted subgraphs of regular graphs

Let G be a 2r-regular, 2r-edge-connected graph of odd order and let m be an integer such that 1 ~ m ~ r 1. For any vertex u of G, the graph G {u} has an m-factor which contains none of r m given edges. All graphs considered are finite. We shall allow graphs to contain multiple edges and we refer the reader to [2] for graph theoretic terms not defined in this paper. Let G be a graph. We denote by K( G) the connectivity of G, which is defined to be the minimum number of vertices whose removal disconnects G or reduces it to ]{l. If G is disconnected K(G) = 0 and if G ~ ]{n, K(G) = n 1, since we must remove n 1 vertices to reduce G to ]{l' G is said to be k-connected if K( G) ~ k. We denote by A( G) the edge connectivity of G, which is defined to be the minimum number of edges whose removal disconnects G. Clearly if G is disconnected A(G) = 0 and if G ~ Knl A(I<n) = n 1. G is said to be k-edge-connected if A( G) ~ k. If S, T ~ V( G) then ea(S, T) denotes the number and Ea(S, T) the set of edges having one end-vertex in S and the other in T. In the case when S is the vertex-set of a subgraph H of G, sometimes we write ea(H, T) and Ea(H, T) instead of ea(V(H), T) and Ea(V(H), T). The number of components of G is denoted by w( G). The minimum degree of the vertices of G is denoted by 8( G). If u E V(G), Na(u) denotes the set of vertices adjacent to u. A k-factor is a k-regular spanning sub graph of G. Thus a Hamilton cycle of a graph is a connected 2-factor and a perfect matching of a graph is a I-factor. Australasian Journal of Combinatorics 15(1997), pp.203-212 The next theorem, due to Petersen, is chronologically the first result on the existence of I-factors. Theorem 1: (Petersen [6]). Every 3-regular, 2-edge-connected graph has a I-factor. In 1934, Schonberger proved the following more general result. Theorem 2: (Schonberger [9]). Let G be a 3-regular, 2-edge-connected graph. Then there exists a I-factor of G that contains no two given edges of G. We now mention two theorems which are generalizations of Schonberger's result. Theorem 3: (Plesnik [7]). Every k-regular, (k -I)-edge-connected graph of even order has a I-factor that contains no k 1 given edges. Theorem 4: (Katerinis [4]). Let G be a k-regular, (k I)-edge connected graph of even order and let m be an integer such that 1 ~ m ~ k 1. Then there exists an m-factor of G that contains no k m given edges of G. The next theorem examines the existence of a I-factor in vertex-deleted subgraphs of a regular graph. Theorem 5:((1): Grant et al. [3], (2): Plesnik [8]) Let G be a 2r-edge-connected, 2r-regular graph of odd order and let u be any vertex of G. Then ( 1) G {u} has a I-factor (2) G {u} has a I-factor which contains none of r 1 given edges. A few years ago, we generalized Theorem 5( 1) in the following way: Theorem 6:(Katerinis [5]). Let G be a 2r-regular, 2r-edge-connected graph of odd order and m be an integer such that 1 ~ m ~ r. Then for every u E V( G), the graph G {u} has an m-factor. The purpose of this paper is to prove the following result which is a generalization of Theorem 5(2).