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9010 results in Discrete Mathematics, Information Theory and Coding

Page 421 of 451

  • A Problem of Erdős and Kátai

  • R. R. Hall
  • Journal:
    Mathematika / Volume 21 / Issue 1 / June 1974
    Published online by Cambridge University Press:
    26 February 2010, pp. 110-113
    Print publication:
    June 1974

  • In this paper I prove the following result:

    Theorem. Let f(d) be a multiplicative function such that |f(d)| ≤ 1 and ∑{ l/p : p ε P} = ∞, where P denotes the set of primes p for which f{p) = −1, and let v(n) denote the number of distinct prime factors ofn. Then for almost all n,

    where A is an arbitrary constant > 3/e.

  • On asymmetry classes of convex bodies

  • Part of
  • Rolf Schneider
  • Journal:
    Mathematika / Volume 21 / Issue 1 / June 1974
    Published online by Cambridge University Press:
    26 February 2010, pp. 12-18
    Print publication:
    June 1974

  • Asymmetry classes of convex bodies have been introduced and investigated by G. Ewald and G. C. Shephard [2], [3], [6]. These classes are defined as follows. Let denote the set of all convex bodies in n-dimensional Euclidean space ℝn. For K1, K2 ∊ write K1 ∼ K2 if there exist centrally symmetric convex bodies S1, S2 ∊ such that

    where + denotes Minkowski addition. Then ∼ is an equivalence relation on and the corresponding classes are called asymmetry classes. The asymmetry class which contains K is denoted by [K].

  • On maximal simplices inscribed in a central convex set

  • Part of
  • James R. McKinney
  • Journal:
    Mathematika / Volume 21 / Issue 1 / June 1974
    Published online by Cambridge University Press:
    26 February 2010, pp. 38-44
    Print publication:
    June 1974

  • Blaschke [1] introduced the notion of maximal tetrahedra inscribed in two and three dimensional convex sets (maximal in the sense of volume). From this notion, he derived an inequality relating the volume of such maximal tetrahedra and the volume of the convex set, and used the inequality to characterize an ellipsoid and to obtain some results concerning isoperimetric inequalities.

  • 5 - Spanning trees and associated structures

  • Norman Biggs, London School of Economics and Political Science
  • Book:
    Algebraic Graph Theory
    Published online:
    05 August 2012
    Print publication:
    16 May 1974, pp 31-37

  • Summary

    The problem of finding bases for the cycle-subspace and the cut-subspace is of great practical and theoretical importance. It was originally solved by Kirchhoff (1847) in his studies of electrical networks, and we shall give a brief exposition of that topic at the end of the chapter.

    We shall restrict our attention to connected graphs, because the cyclesubspace and the cut-subspace of a disconnected graph are the direct sums of the corresponding spaces for the components. Throughout this chapter, Г will denote a connected graph with n vertices and m edges, so that r(Г) = n − 1 and s(Г) = m – n + 1. We shall also assume that Г has been given an orientation.

    A spanning tree in Г is a subgraph which has n − 1 edges and contains no cycles. It follows that a spanning tree is connected. We shall use the symbol T to denote both the spanning tree itself and its edge-set. The following simple lemma is a direct consequence of the definition.

    Lemma 5.1Let T be a spanning tree in a connected graph Г. Then:

    (1) for each edge g of Г which is not in T there is a unique cycle in Г containing g and edges in T only.

    (2) for each edge h of Г which is in T, there is a unique cut in Г containing h and edges not in T only.

  • 11 - The multiplicative expansion

  • Norman Biggs, London School of Economics and Political Science
  • Book:
    Algebraic Graph Theory
    Published online:
    05 August 2012
    Print publication:
    16 May 1974, pp 81-88

  • Summary

    In this chapter and the next one we shall investigate expansions of the chromatic polynomial which involve relatively few subgraphs in comparison with the expansion of Chapter 10. The idea first appeared in the work of Whitney (1932b), and it was developed independently by Tutte (1967) and researchers in theoretical physics who described the method as a ‘linked-cluster expansion’ (Baker 1971). The simple version given here is based on a paper by the present author (Biggs 1973a). There are other approaches which use more algebraic machinery; see Biggs (1978) and lle.

    We begin with some definitions. Recall that if a connected graph Г is separable then it has a certain number of cut-vertices, and the removal of any cut-vertex disconnects the graph. A non-separable subgraph of Г which is non-empty and maximal (considered as a subset of the edges) is known as a block. Every edge is in just one block, and we may think of Г as a set of blocks ‘stuck together’ at the cut-vertices. In the case of a disconnected graph we define the blocks to be the blocks of the components. It is worth remarking that this means that isolated vertices are disregarded, since every block must have at least one edge.

    Let Y be a real-valued function defined for all graphs, and having the following two properties.

    P1: Y(Г) = 1 if Г has no edges;

    P2: Y(Г) is the product of the numbers Y(B) taken over all blocks B of Г.

Page 421 of 451