The Erdős-Sós conjecture for graphs of girth 5

FB Mathematik, Freie Universität Berlin, Graduiertenkolleg ‘Alg. Diskr. Mathematik’, Arnimallee 2–6, 14195 Berlin, Germany; Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
Discrete Mathematics (Impact Factor: 0.58). 01/1996; DOI: 10.1016/0012-365X(95)00207-D

ABSTRACT We prove that every graph of girth at least 5 with minimum degree δ ⩾ k/2 contains every tree with k edges, whose maximum degree does not exceed the maximum degree of the graph. An immediate consequence is that the famous Erdős-Sós Conjecture, saying that every graph of order n with more than n(k − 1)/2 edges contains every tree with k edges, is true for graphs of girth at least 5.

  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: A classical result on extremal graph theory is the Erdös–Gallai theorem: if a graph on n vertices has more than (k-1)n2 edges, then it contains a path of k edges. Motivated by the result, Erdös and Sós conjectured that under the same condition, the graph should contain every tree of k edges. A spider is a rooted tree in which each vertex has degree one or two, except for the root. A leg of a spider is a path from the root to a vertex of degree one. Thus, a path is a spider of 1 or 2 legs. From the motivation, it is natural to consider spiders of 3 legs. In this paper, we prove that if a graph on n vertices has more than (k-1)n2 edges, then it contains every k-edge spider of 3 legs, and also, every k-edge spider with no leg of length more than 4, which strengthens a result of Woźniak on spiders of diameter at most 4.
    Discrete Mathematics 01/2007; 307:3055-3062. · 0.58 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Dobson (1994) conjectured that if G is a graph with girth no less than 2t + 1 and minimum degree no less than k/t and Δ(T), then G contains each tree T of size k.It is known that this conjecture holds for t = 1 and t = 2.We prove it in the case t = 3.
    Discrete Mathematics 01/1997; 165-166:599-605. · 0.58 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The Turán number of a graph H, ex(n, H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let Pl denote a path on l vertices, and let k ⋅ Pl denote k vertex-disjoint copies of Pl. We determine ex(n, k ⋅ P3) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex(n, k ⋅ Pl) for arbitrary l, and n appropriately large relative to k and l. We provide some background on the famous Erdős–Sós conjecture, and conditional on its truth we determine ex(n, H) when H is an equibipartite forest, for appropriately large n.
    Combinatorics Probability and Computing 01/2011; 20:837-853. · 0.61 Impact Factor


1 Download
Available from