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9012 results in Discrete Mathematics, Information Theory and Coding

Page 368 of 451

  • The circumference of a graph with a given minimal degree

  • Edited by A. Baker, Trinity College, Cambridge, B. Bollobás, Trinity College, Cambridge, A. Hajnal
  • Book:
    A Tribute to Paul Erdos
    Published online:
    05 March 2012
    Print publication:
    13 December 1990, pp 97-104

  • Summary

    Abstract

    Extending a theorem of Alon, we prove a conjecture of Katchalski that every graph of order n and minimal degree at least n/k > 1 contains a cycle of length at least n/(k - 1). The result is best possible for all values of n and k (2 ≤ k < n).

    Introduction

    A well-known result of Erdös and Gallai [4] states that, for n ≥ k ≥ 3, a graph of order n and size greater than ½(k - 1)(n - 1) has circumference at least k, that is, it contains a cycle of length at least k. According to Dirac's [3] classical theorem, every graph of order n ≥ 3 and minimal degree at least ½n is Hamiltonian. What can one say about the circumference of a graph of order n and minimal degree at least d ≥ 2? Recently Alon [1] came close to giving a complete answer to this question when he proved that for 2 ≤ k < n every graph of order n and minimal degree at least n/k has circumference at least [n/(k - 1)]. Our aim here is to improve on this slightly, namely to show that the assertion holds without the integer sign, as conjectured by Katchalski. Although this seems to need a surprising amount of work, we feel it is worth it since the new result is best possible for all values of n and k (2 ≤ k < n) and implies a complete answer to the question above concerning the minimal circumference of a graph of order n and minimal degree d ≥ 2.

  • Preface

  • Edited by A. Baker, Trinity College, Cambridge, B. Bollobás, Trinity College, Cambridge, A. Hajnal
  • Book:
    A Tribute to Paul Erdos
    Published online:
    05 March 2012
    Print publication:
    13 December 1990, pp ix-x

  • Summary

    This volume is dedicated to Paul Erdős, who has profoundly influenced mathematics this century. He has worked in number theory, complex analysis, probability theory, geometry, interpolation theory, algebra, set theory and, perhaps above all, in combinatorics. His theorems and conjectures have had a decisive impact. In particular, he, more than anybody else, is the founder of modern combinatorics, he pioneered probabilistic number theory, he is the master of random methods in analysis and combinatorics, and he has created the fields of Ramsey theory and the partition calculus of set theory.

    Paul Erdős is the consummate problem solver: his hallmark is the succinct and clever argument, often leading to a solution from ‘the book’. He loves areas of mathematics which do not require an excessive amount of technical knowledge but give scope for ingenuity and surprise. The mathematics of Paul Erdos is the mathematics of beauty and insight.

    One of the most attractive ways in which Paul Erdős has influenced mathematics is through a host of stimulating problems and conjectures, to many of which he has attached money prizes, in accordance with their notoriety. He often says that he could not pay up if all his problems were solved at once, but neither could the strongest bank if all its customers withdrew their money at the same time. And the latter is far more likely.

Page 368 of 451