On the Existence of General Factors in Regular Graphs

Abstract

Let G be a graph, and H : V (G) → 2 N a set function associated with G. A spanning subgraph F of G is called an H-factor if the degree of any vertex v in F belongs to the set H(v). This paper contains two results on the existence of H-factors in regular graphs. First, we construct an r-regular graph without some given H * -factor. In particular, this gives a negative answer to a problem recently posed by Akbari and Kano. Second, by using Lovász's characterization theorem on the existence of (g, f )-factors, we find a sharp condition for the existence of general H-factors in {r, r + 1}-graphs, in terms of the maximum and minimum of H. The result reduces to Thomassen's theorem for the case that H(v) consists of the same two consecutive integers for all vertices v, and to Tutte's theorem if the graph is regular in addition.

Regular colorings in regular graphs

“…Now we will exhibit r-regular graphs of even order that have (r − 1, 1)colorings but not {r − 1, 1}-factors. The constructions are similar to constructions in [8].…”

“…Since then numerous results on factors have appeared-see, for example, [2,5,7,10]. The concept of factors can be generalized as follows: for any set of integers S, an S-factor of a graph is a spanning subgraph in which the degree of each vertex is in S. Several authors [1,3,8] have recently studied {a, b}-factors in r-regular graphs with a + b = r. In particular, Akbari and Kano [1] made the following conjecture: Conjecture 1.2. If r is odd and 0 ≤ t ≤ r, then every r-regular graph has an {r − t, t}-factor.…”

“…• If t is odd and t ≤ r 2 − 2, then there exists an r-regular graph of even order that has no {r − t, t}-factor ( [8]).…”

Brooks type results for conflict-free colorings and{a,b}-factors in graphs

“…We find a relation between conflict-free colorings and {t, r − t}-factors, studied as early as 1891 by Petersen [20], when the existence of a t-factor in every r-regular graph was proved for r and t even. For each even r = 4 and each odd t there is an r-regular graph without {t, r − t}-factor [16,17]. A 4-regular graph on n vertices has a {1, 3}-factor if and only if n is even.…”

Antifactors of regular bipartite graphs

“…Addario-Berry, Dalal, and Reed [2] slightly improved the result in [1] and obtained a similar result for bipartite graphs. For more results on non-consecutive H-factor problems of graphs, we refer readers to [10,11,15]. However, there is no nice formula to determine whether a bipartite graph contains a 1-anti-factor.…”