Regular factors in regular graphs

Katerinis, P., Regular factors in regular graphs, Discrete Mathematics 113 (1993) 269-274. Let G be a k-regular, (k I)-edge-connected graph with an even number of vertices, and let m be an integer such that 1~ m s k 1. Then the graph obtained by removing any k m edges of G, has an m-factor. All graphs considered are finite. We shall allow graphs to contain multiple edges and we refer the reader tc [l] for standard graph theoretic terms not defined in this paper. Let G be a graph. We say that G has a k-factor, if there exists a k-regular spanning subgraph of G. If S, T c V(G), then ec(S, T) denotes the number and &(S, T) the set of edges having one end-vertex in S and the other in set T. If S c V(G) then w(G S) denotes the number of components of the graph G S. Given an ordered pair (D, S) of disjoint subsets of V(G) and a component C of (G D) -S, put r&D, S; C) = e&V(C), S) + k IV(C)!. We say that C is odd or even component of (G D) S according to whether rc(D, S; C) is odd or even. The number of odd components of (G D) S is denoted by q&D, S; k). Tutte [S] proved the following theorem. Tutte’s k-factor theorem. A graph G has a k-factor if and only if q&D, S; k) + 2 (k dcx(x)) 6 k IDI XES (1) forallD,SgV(G), DnS=8. Correspondence to: P. Katerinis, Department of Informatics, Athens University of Economics and Business, Patission 76, 10434 Athens, Greece. 0012-365X/93/$06.00 CiJ 1993 Elsevier Science Publishers B.V. All rights reserved