Nathan Kahl

  • Citations Per Year
Learn More
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)
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 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, we(More)
We identify best monotone degree bounds for the chromatic number and independence number of a graph. These bounds are best in the same sense as Chvátal’s hamiltonian degree condition. 1 Terminology and Notation We consider only undirected graphs without loops or multiple edges. Our terminology and notation will be standard except as indicated, and a good(More)
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)
Let Γ(n, m) denote the class of all graphs and multigraphs with n nodes and m edges. A central question in network reliability theory is the network augmentation problem: For G ∈ Γ(n, m) fixed, what H ∈ Γ(n, m + k) such that G ⊂ H is t-optimal, that is, maximizes the tree number t(H)? In the network synthesis problem, where G is the empty graph on n(More)
Let ω0(G) denote the number of odd components of a graph G. The deficiency of G is defined as def(G) = maxX⊆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|. Recently(More)
Let τ(G) and τG(e) denote the number of spanning trees of a graph G and the number of spanning trees of G containing edge e of G, respectively. Ferrara, Gould, and Suffel asked if, for every rational 0 < p/q < 1 there existed a graph G with edge e ∈ E(G) such that τG(e)/τ(G) = p/q. In this note we provide constructions that show this is indeed the case.(More)