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- Douglas Bauer, Hajo Broersma, Nathan Kahl, Aurora Morgana, Edward F. Schmeichel, Thomas M. Surowiec
- Discrete Applied Mathematics
- 2007

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)

- Daniel Gross, Nathan Kahl, John T. Saccoman
- Discrete Mathematics
- 2010

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)

- Douglas Bauer, Hajo Broersma, Jan van den Heuvel, Nathan Kahl, Edward F. Schmeichel
- Journal of Graph Theory
- 2013

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)

- Douglas Bauer, Shabnam Hakimi, Nathan Kahl, Edward F. Schmeichel
- Networks
- 2009

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.

- Douglas Bauer, Nathan Kahl, Linda E. McGuire, Edward F. Schmeichel
- Ars Comb.
- 2008

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)

- Nathan Kahl
- 2008

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)

- Arthur H. Busch, Michael Ferrara, Nathan Kahl
- Discrete Applied Mathematics
- 2007

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)

- Nathan Kahl
- Discrete Applied Mathematics
- 2016

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)

- Douglas Bauer, M. Yatauro, Nathan Kahl, Edward F. Schmeichel
- Discrete Mathematics
- 2011

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.