The linear complementarity problem: find z∈Rp satisfying is generalized to a problem in which the matrix M is not square. A solution technique similar to C. E. Lemke's (1965) method for solving (LCP) is given. The method is discussed from a graph-theoretic viewpoint and closely parallels a proof of Sperner's lemma by D. I. A. Cohen (1967) and some work of H. Scarf (1967) on approximating fixed points of a continuous mapping of a simplex into itself.
It is shown in this paper that a map not colorable in four colors must have at least n=40 countries. This improves on the result n=36 due to C. E. Winn (1940). The rather elaborate computations are based upon the Euler contributions of the faces in an irreducible graph and upon several new reducible configurations.
This review article presents three methods for solving enumeration problems which can be construed as the determination of the number of orbits of an appropriate permutation group. Such a group must be constructed in accordance with the idiosyncrasies of the configurations to be counted and the equivalence relation on them. Thus three different binary operations on permutation groups, the sum, product, and power group, are defined and the structure of each is investigated. Applications of the corresponding theorems for enumeration under group action are provided. We conclude with a table of 27 current unsolved problems in graphical enumeration.
Let the vertices of an undirected graph be given labels 1, 2, …, n, 1′, 2′, …, n′ such that each vertex has at least n−1 different labels without both i and i′ for any i. Then among all paths between a vertex labeled i and a vertex labeled i′ for any i, the maximum number which are mutually edge disjoint equals the minimum size of an edge cut-set separating all vertices labeled i from all those labeled i′ for any i.
Let γ denote the number of points in the path joining two arbitrary points in a random tree Tn with n labeled points. It is shown, among other things, that E(γ)∼(1/2πn)1/2.
An algebraic structure, related to the crossing number, is constructed from a given graph.
Many of the basic theorems about general “algebras” derived in [1, Ch. 6] are extended to a class of heterogeneous algebras which includes automata, state machines, and monoids acting on sets. It is shown that some algebras can be fruitfully studied, using different interpretations, both as (homogeneous) algebras and as heterogeneous algebras, and a non-trivial “free machine” is constructed as an application. The extent of the overlap with previous work of Higgins [9] is specified.
The conditions imposed by edge-transitivity and vertex-transitivity on the connectivity of simple graphs are investigated. Particular attention is given to the structure of those vertex-transitive graphs for which the degree of regularity exceeds the connectivity.
A theorem is established that provides necessary and sufficient conditions in order that a locally finite bipartite graph have a subgraph whose valences lie in prescribed intervals. This theorem is applied to the study of flows in locally finite directed graphs. In particular, generalizations of the max-flow min-cut theorem and of the circulation theorem are obtained.
The axiom of choice is assumed throughout.
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