Article
The ErdősSós 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(13):411414. DOI: 10.1016/0012365X(95)00207D 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 ErdősSós 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.
Fulltext preview
sciencedirect.com Available from: Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.

 "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 . "
[Show abstract] [Hide abstract]
ABSTRACT: The ErdősSós 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. 
 "Brandt and Dobson [10] "
[Show abstract] [Hide abstract]
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. 
 "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]. "
Article: The Erdös–Sós conjecture for spiders
[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 (k1)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 (k1)n2 edges, then it contains every kedge spider of 3 legs, and also, every kedge spider with no leg of length more than 4, which strengthens a result of Woźniak on spiders of diameter at most 4.