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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)

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)

One of Frank Boesch's best known papers is 'The strongest monotone degree conditions for n-connectedness of a graph' [1]. 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.

Dirac showed that a 2–connected graph of order n with minimum degree δ has circumference at least min{2δ, n}. We prove that a 2– connected, triangle-free graph G of order n with minimum degree δ either has circumference at least min{4δ−4, n}, or every longest cycle in G is dominating. This result is best possible in the sense that there exist bipartite… (More)

The independence polynomial I(G; x) of a graph G is I(G; x) = α(G) k=0 s k x k , where s k is the number of independent sets in G of size k. The decycling number of a graph G, denoted φ(G), is the minimum size of a set S ⊆ V (G) such that G − S is acyclic. Engström proved that the independence polynomial satisfies |I(G; −1)| ≤ 2 φ(G) for any graph G, and… (More)