Even [a,b]-factors in graphs

Abstract

Let a and b be integers 4 ≤ a ≤ b. We give simple, sufficient conditions for graphs to contain an even [a, b]-factor. The conditions are on the order and on the minimum degree, or on the edge-connectivity of the graph.

More on even [a,b]-factors in graphs

“…Kouider and Vestergaard [3] apply Lovász' parity [g, f ]-factor theorem (Theorem 5 below) to find simple sufficient conditions for graphs to contain even [a, b]-factors. In this section we also apply Theorem 5 to obtain different sufficient conditions for graphs to have even [a, b]-factors.…”

“…Let G be a graph with vertex set V (G) and edge set E(G). Denote by deg G (v) the degree of a vertex v in G and δ(G) the minimum degree of G. Let (X, Y ) be an ordered pair of distinct subsets of V (G) Theorem 1 ( [2,3]). Let a and b be even integers such that 2 ≤ a ≤ b and let G be a connected graph of order n.…”

“…In addition, Kouider and Vestergaard [2] obtained a characterization of the complete bipartite graphs which have an even [2, b]-factor. The reader may consult [4] for a recent survey on connected factors in graphs.…”

“…Then we give a characterization of the complete bipartite graphs which have an odd [a, b]-factor. For general graphs, we modify the proof of Theorem 2 of [2] to prove that an a-edge connected graph G with n vertices and with δ(G) ≥ max{a + 1, an a+b + a − 2} has an even [a, b]-factor, where a and b are even and 2 ≤ a ≤ b. The edge-connectivity condition required in our result is slightly weaker than the one given in Part (3) of Theorem 1.…”

Connected Factors in Graphs ? a Survey

“…One variant is that the parity of the degrees may be prescribed, see e.g. [131], [38], [2] for odd factors, [89] for even factors and [35] for both. Factors may also be selected by some structural property, and we shall, in this survey, do exactly that by putting our emphasis on connected factors.…”