Stephen B. Maurer

Matroid basis graphs. II

Abstract Several graph-theoretic notions applied to matroid basis graphs in the preceding paper are now tied more specifically to aspects of matroids themselves. Factorizations of basis graphs and disconnections of neighborhood subgraphs are related to matroid separations. Matroids are characterized whose basis graphs have only one or two of the three types of common neighbor subgraphs. The notion of leveling is generalized and related to matroid sums, minors, and duals. Also, the problem of characterizing regular and graphic matroids through their basis graphs is discussed. Throughout, many results are obtained quite easily with the aid of certain pseudo-combivalence systems of 0–1 matrices.

On k-critical, n-connected graphs☆

A graph G which is n-connected (but not (n + 1)-connected)is defined to be k-critical if for every S ⊆ V(G), where |S|⩽k, the connectivity of G − S is n − |S|. We will say that G is an (n∗,k∗) graph if G is n-connected (but not (n + 1)-connected) and k-critical (but not (k + 1)- critical). This initial study of k-critical graphs is concerned with the problem of determining the values of n and k for which there exists an (n∗, k∗) graph.

Matrix Generalizations of Some Theorems on Trees, Cycles and Cocycles in Graphs

We extend to arbitrary matrices four theorems of graph theory, one about projections onto the cycle and cocycle spaces, one about the intersection of these spaces, and two matrix-tree theorems. The squares of certain determinants, not apparent for graphs, appear in the extensions.

On induced economic change in precapitalist societies

Abstract This essay explores a number of properties of a general growth model of induced economic change in precapitalist societies which incorporates both the insights of Ester Boserup and Thomas Malthus. Responses to diminishing returns include changes in work intensity and population growth. Some important variants of the model are also examined which focus on the transfer of production to non-producers and on alternative processes by which change is induced. The results are used to generate some parameters influencing the political stability of non-working elites; to criticize some previous, less general, growth models; and to suggest some fruitful lines fot future empirical research on economic development in the long run.

Vertex colorings without isolates

Abstract Call a vertex of a vertex-colored simple graph isolated if all its neighbors have colors other than its own. A. J. Goldman has asked: When is it possible to color b vertices of a graph black and the remaining w vertices white so that no vertex is isolated? We prove (1) if G is connected and has minimum degree 2, it is always possible unless b or w is 1; (2) if G is 2-connected, then for any pair ( b , w ) there is a coloring in which both monochromatic subgraphs are connected; (3) if G has vertices of degree 1, a necessary condition for a ( b , w ) coloring without isolates to exist is that there be a solution to a certain knapsack inequality. Next, statements generalizing (1) and (2) to n colors are presented, and current knowledge about their truth is discussed. Then various refinements of (1) and (3), more complicated to state and prove, are given. For instance, with the hypotheses of (1) at least one of the monochromatic subgraphs may be chosen to be connected. Also, the necessary knapsack inequality of (3) is, in most cases, sufficient. Throughout, some consideration is given to the algorithmic complexity of coloring (if possible) without isolates. For most graphs which might arise in practice there is an efficient algorithm for the 2-color problem. However, for arbitrary graphs the 2-(or more) color problem is NP-complete.

On k-minimally n-edge-connected graphs

Abstract A graph is k -minimal with respect to some parameter if the removal of any j edges j k reduces the value of that parameter by j . For k = 1 this concept is well-known; we consider multiple minimality, that is, k ⩾ 2. We characterize all graphs which are multiply minimal with respect to connectivity or edge-connectivity. We also show that there are essentially no diagraphs which are multiply minimal with respect to diconnectivity or edge-diconnectivity. In addition, we investigate basic properties and multiple minimality for a variant of edge-connectivity which we call edge m -connectivity.

Large minimal realizers of a partial order II

The rank (resp. dimension) of a poset P is the cardinality of a largest (resp. smallest) set of linear orders such that its intersection is P and no proper subset has intersection P. Dimension has been studied extensively. Rank was introduced recently by Maurer and Rabinovitch in [4], where the rank of antichains was determined. In this paper we develop a general theory of rank. The main result, loosely stated, is that to each poset P there corresponds a class of graphs with easily described properties, and that the rank of P is the maximum number of edges in a graph in this class.

Advice For Undergraduates On Special Aspects Of Writing Mathematics

ABSTRACT There are several guides to good mathematical writing for professionals, but few for undergraduates. Yet undergraduates who write mathematics papers need special guidance. For instance, professionals may need help writing clear definitions, but at least they know why explicit definitions are needed and know the basic format. In general, mathematics has many special formats that are not mere technical conventions but instead serve important purposes. Often students are not conscious of these conventions, and they rarely know their purposes. This paper contains a guide for students written in light of these observations.

Taxes and capital intensity in a two-class disposable income growth model

Abstract By analyzing the incentive effects of taxes, previous studies may give the impression that the impact on capital intensity of converting an income tax to either a consumption or wage tax depends solely on the difference in such incentive effects on the representative person. This paper shows that tax conversion may also alter capital intensity by shifting disposable income from low to high savers. In a two-class, disposable income growth model that eliminates the incentive effects of taxes, converting an income tax to a wage or consumption tax that would raise equal steady-state revenue per worker would raise the steady-state capital intensity of the economy.

The Effects of a New College Mathematics Curriculum on High School Mathematics

The title is too narrow by half. It suggests the influence will go all one way, with changes at the college level filtering down to the secondary. Perhaps in the long run that will be true, but in the short run, to the extent that the interface between the two levels is now carefully synchronized, what is presently taught at one level constrains what changes can be made unilaterally at the other. Even in the long run, if changes at the college level demand major changes at the secondary level for which secondary teachers are not ready because of training and philosophy, then changes at the secondary level will be very slow.