A>factors and Neighbourhoods of Independent Sets in Graphs

Sufficient conditions are proved for the existence of a A>factor or a Hamiltonian circuit in a graph G. The main element of each condition is a statement to the effect that \N(X)\ > a\X\ + b\V(G)\ + c for every non-empty independent subset X of V(G). The theorems are shown to contain various known results. 1. The theorems Neighbourhood conditions in graphs have been much studied [1, 2, 4, 6, 10, 11], but only recently has attention focused on independent sets. The present paper arose from the observation that various results proved in the literature do not use the full force of their hypotheses. We prove the new results in this section, and show in the next that they contain the old ones. Recall that if G is a graph with vertex set V(G) and X £ V(G), then N(X):= vex where N(v) is the set of vertices adjacent to ('neighbouring') v. We call a graph G sesquiconnected if it is connected and, for each vertex v in G, G\{v) has at most two components. THEOREM 1. Let G be a sesquiconnected graph with n vertices (n even) and minimum degree S ^ \(n + 3) such that 5) (1) for every non-empty independent subset X of V(G). Then G has a \-factor. Proof. If it does not, then by Tutte's characterization [8] G contains a set S of vertices such that G\S has at least 151 + 2 odd components. Each of these must have at least \(n + 3) + 1 — |S| vertices since S ^ \(n + 3), and so