Nathan Kahl

Best Monotone Degree Conditions for Graph Properties: A Survey

We survey sufficient degree conditions, for a variety of graph properties, that are best possible in the same sense that Chvátal’s well-known degree condition for hamiltonicity is best possible.

Toughness and Vertex Degrees

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 ti¾?1, but then show that for any integer ki¾?1, a best monotone theorem for t=1ki¾?1 requires at least fki¾?|VG| nonredundant conditions, where fk grows superpolynomially as ki¾?∞. When t<1, we give an additional, simple theorem for G to be t-tough, in terms of its vertex degrees.

Tutte sets in graphs II: The complexity of finding maximum Tutte sets

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 of Tutte sets in another paper. Here, we consider the algorithmic complexity of finding Tutte sets in a graph. We first give two polynomial algorithms for finding a maximal Tutte set. We then consider the complexity of finding a maximum Tutte set, and show it is NP-hard for general graphs, as well as for several interesting restricted classes such as planar graphs. By contrast, we show we can find maximum Tutte sets in polynomial time for graphs of level 0 or 1, elementary graphs, and 1-tough graphs.

Toughness and binding number

Let @t(G) and bind(G) be the toughness and binding number, respectively, of a graph G. Woodall observed in 1973 that @t(G)>=bind(G)-1. In this paper, we obtain best possible improvements of this inequality except when (1+5)/2

Best monotone degree conditions for binding number

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 Chvatals well-known degree condition to guarantee a graph is Hamiltonian is best possible.

A note on the values of independence polynomials at - 1

The independence polynomial I ( G ; x ) of a graph G is I ( G ; x ) = ź k = 0 α ( G ) 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 ź ( G ) , is the minimum size of a set S ź V ( G ) such that G - S is acyclic. Engstrom proved that the independence polynomial satisfies | I ( G ; - 1 ) | ź 2 ź ( G ) for any graph G , and this bound is best possible. Levit and Mandrescu provided an elementary proof of the bound, and in addition conjectured that for every positive integer k and integer q with | q | ź 2 k , there is a connected graph G with ź ( G ) = k and I ( G ; - 1 ) = q . In this note, we prove this conjecture.

On constructing rational spanning tree edge densities

Let ź ( G ) and ź G ( 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 ź E ( G ) such that ź G ( e ) / ź ( G ) = p / q . In this note we provide constructions that show this is indeed the case. Moreover, we show this is true even if we restrict G to claw-free graphs, bipartite graphs, or planar graphs. Let dep ( G ) = max e ź G ź G ( e ) / ź ( G ) . Ferrara et al. also asked if, for every rational 0 < p / q < 1 there existed a graph G with dep ( G ) = p / q . For the claw-free construction, we are also able to answer this question in the affirmative.