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Papers by Edward Schmeichel

Annals of Discrete Mathematics, 1993

Ars Combinatoria Waterloo Then Winnipeg, 1998

Discrete Applied Mathematics, 2002

A number of results in hamiltonian graph theory are of the form P 1 implies P 2 , where P 1 is a ... more A number of results in hamiltonian graph theory are of the form P 1 implies P 2 , where P 1 is a property of graphs that is NP-hard and P 2 is a cycle structure property of graphs that is also NP-hard. Such a theorem is the well-known Chvátal-Erdös Theorem, which states that every graph G with α ≤ κ is hamiltonian. Here κ is the vertex connectivity of G and α is the cardinality of a largest set of independent vertices of G. In another paper Chvátal points out that the proof of this result is in fact a polynomial time construction that either produces a Hamilton cycle or a set of more than κ independent vertices. In this note we point out that other theorems in hamiltonian graph theory have a similar character. In particular, we present a constructive proof of the well-known theorem of Jung [12] for graphs on 16 or more vertices.

We consider sufficient conditions for a degree sequence pi\pipi to be forcibly kkk-factor graphical... more We consider sufficient conditions for a degree sequence pi\pipi to be forcibly kkk-factor graphical. We note that previous work on degrees and factors has focused primarily on finding conditions for a degree sequence to be potentially kkk-factor graphical. We first give a theorem for pi\pipi to be forcibly 1-factor graphical and, more generally, forcibly graphical with deficiency at most betage0\beta\ge0betage0.

Israel Journal of Mathematics

Ars Combinatoria -Waterloo then Winnipeg-

Let t # 1 be an integer. We show that it is NP-hard to determine if an r-regular graph is t-tough... more Let t # 1 be an integer. We show that it is NP-hard to determine if an r-regular graph is t-tough for any fixed integer r # 3 t.Wealso discuss the complexity of recognizing if an r-regular graph is t-tough, for any rational t # 1. Keywords : toughness, regular graph, NP-completeness AMS Subject Classifications (1991) : 68R10, 05C38 # Supported in part by the National Science Foundation under Grant DMS9206991. 1 1 Introduction We begin with a few definitions and some notation. A good reference for any undefined terms in graph theory is [7] and for computational complexity is [12]. We consider only undirected graphs with no loops or multiple edges. Let G be a graph. Then G is hamiltonian if it has a Hamilton cycle, i.e., a cycle containing all of its vertices. It is traceable if it has a path containing all of its vertices. Let #(G) denote the number of components of G.We say G is t-tough if |S|#t #(G - S) for every subset S of the vertex set V (G)ofG with #(G - S) > 1...

We identify best monotone degree bounds for the chromatic number of independence number of a grap... more We identify best monotone degree bounds for the chromatic number of independence number of a graph. These bounds are best in the same sense as Chvátal’s Hamiltonian degree condition.

We survey sufficient degree conditions, for a variety of graph properties, that are best possible... more 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.

Mathematics Magazine, 1997

[1988] The Eighteenth International Symposium on Fault-Tolerant Computing. Digest of Papers, 1988

... Bleclier [2] proposed an adaptive algoritliin wliicli re-quired n + t - 1 tests and n + t - 1... more ... Bleclier [2] proposed an adaptive algoritliin wliicli re-quired n + t - 1 tests and n + t - 1 rouiids to identify the faulty units; Bleclier also gave ail adversary argu-iiieiit to establish that any adaptive algoritliiii requires n + t - 1 tests in the worst case to identify the faulty uiii t s . In 171 ...

Israel Journal of Mathematics, 1982

LetG be ap-vertex planar graph having a representation in the plane with nontriangular facesF 1,F... more LetG be ap-vertex planar graph having a representation in the plane with nontriangular facesF 1,F 2, …,F r. Letf 1,f 2, …,f r denote the lengths of the cycles bounding the facesF 1,F 2, …,F r respectively. LetC 3(G) be the number of cycles of length three inG. We give bounds onC 3(G) in terms ofp,f 1,f 2, …,f r.