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Let ω0(G) denote the number of odd components of a graph G. The deficiency of G is defined as def (G) = max X⊆V (G) (ω0(G − X) − |X|), and this equals the number of vertices unmatched by any maximum matching of G. A subset X ⊆ V (G) is called a Tutte set (or barrier set) of G if def (G) = ω0(G − X) − |X|, and an extreme set if def (G − X) = def (G) + |X|.… (More)
We study theorems giving sufficient conditions on the vertex degrees of a graph G to guarantee G is t-tough. We first give a best monotone theorem when t ≥ 1, but then show that for any integer k ≥ 1, a best monotone theorem for t = 1 k ≤ 1 requires at least f (k) · |V (G)| nonredundant conditions, where f (k) grows superpolynomially as k → ∞. When t < 1,… (More)
Recently Alon and Friedland have shown that graphs which are the union of complete regular bipartite graphs have the maximum number of 1-factors over all graphs with the same degree sequence. We identify two families of graphs that have the maximum number of 1-factors over all graphs with the same number of vertices and edges: the almost regular graphs… (More)
One of Frank Boesch's best known papers is 'The strongest monotone degree conditions for n-connectedness of a graph' . In this paper, we give a simple sufficient degree condition for a graph to be k-edge-connected, and also give the strongest monotone condition for a graph to be 2-edge-connected.
We consider sufficient conditions for a degree sequence π to be forcibly k-factor graphical. We note that previous work on degrees and factors has focused primarily on finding conditions for a degree sequence to be potentially k-factor graph-ical. We first give a theorem for π to be forcibly 1-factor graphical and, more generally , forcibly graphical with… (More)
We give sufficient conditions on the vertex degrees of a graph G to guarantee that G has binding number at least b, for any given b > 0. Our conditions are best possible in exactly the same way that Chvátal's well-known degree condition to guarantee a graph is hamiltonian is best possible.