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9084 results in Discrete Mathematics, Information Theory and Coding

Page 328 of 455

  • On the Maximum Number of Triangles in Wheel-Free Graphs

    • By Z. Füredi, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, M.X. Goemans, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, D.J. Kleitman, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139
  • Edited by Béla Bollobás, University of Cambridge, Andrew Thomason, University of Cambridge
  • Book:
    Combinatorics, Geometry and Probability
    Published online:
    06 December 2010
    Print publication:
    22 May 1997, pp 305-318

  • Summary

    Gallai raised the question of determining t(n), the maximum number of triangles in graphs of n vertices with acyclic neighborhoods. Here we disprove his conjecture (t(n) ∼ n2/8) by exhibiting graphs having n2/7.5 triangles. We improve the upper bound of (n2 − n)/6 to t(n) ≤ n2/7.02 + O(n). For regular graphs, we further decrease this bound to n2/7.75 + O(n).

    Introduction

    Let WFGn be the class of graphs on n vertices with the property that the neighborhood of any vertex is acyclic. A graph G is given by its vertex set V(G) and edge set E(G). The subgraph induced by X ⊂ V(G) is denoted by G[X]. The neighborhood N(v) of vertex v is the set of vertices adjacent to v. Note that v ∉ N(v). The degree of v ∈ V(G), denoted by dv or dv(G), is the size of the neighborhood: dv = |N(v)|. The maximum (minimum) degree is denoted by Δ (δ), or Δ(G) (δ(G), respectively) to avoid misunderstandings. A matching M ⊂ E(G) is a set of pairwise disjoint edges. A wheel Wi is obtained from a cycle Ci by adding a new vertex and edges joining it to all the vertices of the cycle; the new edges are called the spokes of the wheel (i ≥ 3, W3 = K4). Therefore, WFGn consists of all graphs on n vertices containing no wheel.

  • (1,2)-Factorizations of General Eulerian Nearly Regular Graphs

    • By R. Häggkvist, Department of Mathematics, University of Umeá, S-901 87 Umeá, Sweden, A. Johansson, Department of Mathematics, University of Umeá, S-901 87 Umeá, Sweden
  • Edited by Béla Bollobás, University of Cambridge, Andrew Thomason, University of Cambridge
  • Book:
    Combinatorics, Geometry and Probability
    Published online:
    06 December 2010
    Print publication:
    22 May 1997, pp 329-338

  • Summary

    Every general graph with degrees 2k and 2k − 2,k ≥ 3, with zero or at least two vertices of degree 2k − 2 in each component, has a k-edge-colouring such that each monochromatic subgraph has degree 1 or 2 at every vertex.

    In particular, if T is a triangle in a 6-regular general graph, there exists a 2-factorization of G such that each factor uses an edge in T if and only if T is non-separating.

    Introduction

    In this paper we will characterize those general graphs with degrees 2k − 2 and 2k that can be decomposed into spanning subgraphs with degrees 1 and 2 everywhere. Before we state the result, it is perhaps of some interest to review some related problems and their history.

    Background

    One of the starting points of graph theory is a classic investigation by the Danish mathematician Julius Petersen who in 1891 published a paper: ‘Die Theorie der regulären graphs’, which contains a wealth of material on the problem of factorizing regular graphs into graphs of uniform degree k. An excellent source of information concerning Julius Petersen and problems spawned by his 1891 paper is the conference volume.

    The motivation for Petersen's work, as given in the first few lines of his article, came from Hilbert's proof of the finiteness of the system of invariants associated with a binary form.

  • Clique Partitions of Chordal Graphs

    • By Paul Erdős, Mathematical Institute, Hungarian Academy of Sciences, E.T. Ordman, Memphis State University, Memphis, TN 38152 U.S.A, Y. Zalcstein, Division of Computer and Computation Research, National Science Foundation, Washington, D. C. 20550, U. S. A
  • Edited by Béla Bollobás, University of Cambridge, Andrew Thomason, University of Cambridge
  • Book:
    Combinatorics, Geometry and Probability
    Published online:
    06 December 2010
    Print publication:
    22 May 1997, pp 291-298

  • Summary

    To partition the edges of a chordal graph on n vertices into cliques may require as many as n2/6 cliques; there is an example requiring this many, which is also a threshold graph and a split graph. It is unknown whether this many cliques will always suffice. We are able to show that (1 − c)n2/4 cliques will suffice for some c > 0.

    Introduction

    We consider undirected graphs without loops or multiple edges. The graph Kn on n vertices for which every pair of distinct vertices induces an edge is called a complete graph or a clique on n vertices. If G is any graph, we call any complete subgraph of G a clique of G (we do not require that it be a maximal complete subgraph). A clique covering of G is a set of cliques of G that together contain each edge of G at least once; if each edge is covered exactly once we call it a clique partition. The clique covering number cc(G) and clique partition number cp(G) are the smallest cardinalities of, respectively, a clique covering and a clique partition of G.

    The question of calculating these numbers was raised by Orlin in 1977. DeBruijn and Erdős had already proved, in 1948, that partitioning Kn into smaller cliques required at least n cliques. Some more recent studies motivating the current paper include.

Page 328 of 455