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The Erdős-Sós conjecture for graphs of girth 5

Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
Discrete Mathematics (Impact Factor: 0.56). 04/1996; 150(1-3):411-414. 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.

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    • "In the proof of Theorem 1.1 we follow the terminology introduced in [13] [14] which we recall next. A vertex of degree 1 is a leaf, and a penultimate vertex in a tree is a leaf in the subtree of T obtained by deleting all leaves of T . "
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    ABSTRACT: The Erdős-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k− 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ζ(G) of a graph G defined as ζ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ζ(G) ≥ 2k − 4 contains every tree of k edges if d G (x) + d G (y) ≥ 2k − 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d G (u) ≥ k.
    Full-text · Article · Jan 2011 · Acta Mathematica Sinica English Series
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    • "Brandt and Dobson [10] "
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    ABSTRACT: Erdös and Sós conjectured in 1963 that every graph G on n vertices with edge number e(G) > ½ (k − 1)n contains every tree T with k edges as a subgraph. In this paper, we consider a variation of the above conjecture, that is, for n ≥ 9/2 k 2 + 37/2 k+14 and every graph G on n vertices with e(G) > ½ (k − 1)n, we prove that there exists a graph G′ on n vertices having the same degree sequence as G and containing every tree T with k edges as a subgraph.
    Preview · Article · May 2009 · Acta Mathematica Sinica
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    • "There are only few partial results known, mainly on two directions. One is to pose conditions on the graph G, such as graphs of girth 5 by Brandt and Dobson [1], and then improved to graphs without cycle of length 4 by Saclé and Wo´zniak [4]. The other is to pose conditions on the tree, such as trees with a vertex joined to at least k−1 2 vertices of degree 1 by Sidorenko [5], and spiders of diameter at most 4 by Wo´zniak [6]. "
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    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.
    Preview · Article · Nov 2007 · Discrete Mathematics
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