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A well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph G in terms of what is usually called the deficiency. A subset X of V (G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. We explored the structural aspects(More)
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)
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)