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MAXIMAL NONHAMILTONIAN GRAPHS, STABILITY AND TOTAL VERTICES

We consider only finite undirected graphs without loops or multiple edges. For the sake of completeness we recall some definitions. Let G=(V,E) be a graph with the vertex set V=V(G) and the edge set E=E(G). U(G) denotes the number of components of G. The graph G is tough if |S|SU(G\S) for any ScV with (j(G\S)>1. We shall denote by a=a(G) the cardinality of a maximum set of independent vertices of G and by K(G) the connectivity of G. A vertex veV(G) is called total iff v is adjacent to all remaining vertices of G. A complete graph with n vertices is denoted by as usually. Given graphs G and H, HcG means that H is a subgraph of G, i.e. V(H)cV(G) and E(H)cE(G). If at the same time V(H)=V(G) then H is a factor of G. The star * denotes the operation of join on vertex disjoint graphs, with the convention that F*G*H = (F*G)u(G*H), where u denotes the ordinary union of (not necessarily disjoint) graphs. GuH stands for vertex disjoint union of the graphs G and H.