Michael D. Plummer

On n-extendable graphs

A graph G is said to be n-extendable if it is connected, has a set of n independent lines and every set of n independent lines in G extends to (i.e. is a subset of) a perfect matching of G. Nearly all n-extendable graphs (n>=2) are shown to be (n-1)-extendable and (n + 1)- connected. The special cases of 2- and 3-extendable graphs and their relationships with bicritical and elementary bipartite graphs are then studied.

WELL-COVERED GRAPHS: A SURVEY

Abstract A graph G is well-covered (or w-c) if every maximal independent set of points in G is also maximum. Clearly, this is equivalent to the property that the greedy algorithm for constructing a maximal independent set always results in a maximum independent set. Although the problem of independence number is well-known to be NP-complete, it is trivially polynomial for well-covered graphs. The concept of well-coveredness was introduced by the author in [P1] and was first discussed therein with respect to its relationship to a number of other properties involving the independence number. Since then, a number of results about well-covered graphs have been obtained. It is our purpose in this paper to survey these results for the first time. As the reader will see, many of the results we will discuss are quite recent and have not as yet appeared in print.

Extending matchings in graphs: a survey

Gallai and Edmonds independently obtained a canonical decomposition of graphs in terms of their maximum matchings. Unfortunately, one of the degenerate cases for their theory occurs when the graph in question has a perfect matching (also known as a 1-factor). Kotzig, Lovasz and others subsequently developed a further decomposition of such graphs. Among the `atoms? of this decomposition is the family of bicritical graphs. (A graph G is bicritical if G - u - v has a perfect matching for every choice of two points u, v in G.) So far graphs have resisted further decomposition procedures.Motivated by these mysterious graphs, we introduced the following definition. Let p and n be positive integers with n </ (p ? 2)/2. Graph G is n-extendable if G has a matching of size n and every such matching extends to (i.e. is a subset of) a perfect matching in G. It is clear that if a graph is bicritical, it is 1-extendable. A more interesting result is that if a graph is 2-extendable, it is either bipartite or bicritical. It is also true that if a graph is n-extendable, it is also (n ? 1)-extendable. Hence, for nonbipartite graphs we have a nested sequence of families of bicritical graphs to study.In this paper, we survey a variety of results obtained over the past few years concerning n-extendable graphs. In particular, we describe how the property of n-extendability interacts with such other graph parameters as genus, toughness, claw-freedom and degree sums and generalized neighborhood conditions. We will also investigate the behavior of matching extendability under the operation of Cartesian product. The study of n-extendability for planar graphs has been, and continues to be, of particular interest.

Perspectives: Graph factors and factorization: 1985-2003: A survey

In the most general sense, a factor of a graph G is just a spanning subgraph of G and a graph factorization of G is a partition of the edges of G into factors. However, as we shall see in the present paper, even this extremely general definition does not capture all the factor and factorization problems that have been studied in graph theory. Several previous survey papers have been written on this subject [F. Chung, R. Graham, Recent results in graph decompositions, London Mathematical Society, Lecture Note Series, vol. 52, Cambridge University Press, 1981, pp. 103-123; J. Akiyama, M. Kano, Factors and factorizations of graphs-a survey, J. Graph Theory 9 (1985) 1-42; L. Volkmann, Regular graphs, regular factors, and the impact of Petersens theorems, Jahresber. Deutsch. Math.-Verein. 97 (1995) 19-42] as well as an entire book on graph decompositions [J. Bosak, Decompositions of Graphs, Kluwer Academic Publishers Group, Dordrecht, 1990]. Our purpose here is to concentrate primarily on surveying the developments of the last 15-20 years in this exponentially growing field.

Cyclic coloration of 3‐polytopes

Abstract : This paper, all graphs will be finite, loopless and will have no parallel lines. Let G be a 2-connected planar graph with V(G)=p points. Suppose G has some fixed imbedding Phi: G approaches R-sq in the plane. The pair (G Phi) is often called a plane graph. A cyclic coloration of (G Phi) is an assignment to colors to the points of G such that for any face-bounding cycle F of (G Phi), the points of F have different colors. The cyclic coloration number chi sub c ((G Phi)) is the minimum number of colors in any cyclic coloration of (G, Phi). The main result of the present paper is to show that if (G, Phi) is a 3-connected plane graph, then chi sub c (G, Phi) p* (G, Phi)+ 9. Moreover, if rho* is sufficiently large of sufficiently large or sufficiently small, then this bound on chi sub c can be improved somewhat.

A nine point theorem for 3-connected graphs

We prove that a 3-connected cubic graph contains a cycle through any nine points.

A Theorem on Matchings in the Plane

It is shown that any connected planar graph G with at least 8 vertices and having a perfect matching contains a set S of three independent lines with the property that S is not contained in any perfect matching of G .

On a special case of Hadwiger's conjecture

Hadwigers Conjecture seems difficult to attack, even in the very special case of graphs G of independence number α (G) = 2. We present some results in this special case.

Matching extension and the genus of a graph

Abstract : Let G be a graph with p points having a perfect matching and suppose n is a positive integer with n or = (p-2)/2. Then G is n-extendable if every matching in G containing n lines is a subset of a perfect matching. In this paper we obtain an upper bound on the n-extendability of a graph in terms of its genus. Keywords: Euler contributions; Theorems. (Author)

A pair of forbidden subgraphs and perfect matchings

In this paper, we study the relationship between forbidden subgraphs and the existence of a matching. Let H be a set of connected graphs, each of which has three or more vertices. A graph G is said to be H-free if no graph in H is an induced subgraph of G. We completely characterize the set H such that every connected H-free graph of sufficiently large even order has a perfect matching in the following cases.(1) Every graph in H is triangle-free. (2) H consists of two graphs (i.e. a pair of forbidden subgraphs).A matching M in a graph of odd order is said to be a near-perfect matching if every vertex of G but one is incident with an edge of M. We also characterize H such that every H-free graph of sufficiently large odd order has a near-perfect matching in the above cases.