The Erdős-Sós conjecture for graphs of girth 5

Brandt, Stephan; Dobson, Edward

doi:10.1016/0012-365x(95)00207-d

Cited by 52 publications

(37 citation statements)

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“…Our work sheds some light on why the Erdős-Sós conjecture, which we already discussed in the beginning of the introduction, becomes easier for graphs with no short cycles or no complete bipartite subgraphs K 2,r . These scenarios were considered, e.g., in [8,15,12,16]. In particular, assuming that graph G has girth 2k +1,k ≥ 2, and minimum degree d, Jiang [12] showed how to embed in G all trees with kd vertices and degrees bounded by d. Although this is best possible, our result implies that this statement can be tight only for relatively few very special trees, i.e., those that contain several large stars of degree extremely close to d. Indeed, if we relax the degree assumption and consider trees with maximum degree at most (1 − )d, then it is possible to embed trees of order O(d k ) rather than O(d).…”

Section: Discussionmentioning

confidence: 99%

“…The conjecture has been proved in several special cases, e.g. Brandt and Dobson [8] establish it for graphs of girth at least 5 (the girth is the length of the shortest cycle in a graph). In fact, they prove a stronger statement, that any such graph of minimum degree d/2 and maximum degree Δ contains all trees with d edges and maximum degree at most Δ.…”

Section: Extremal Questionsmentioning

confidence: 99%

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A randomized embedding algorithm for trees

Sudakov¹,

Vondrák

2010

Combinatorica

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In this paper, we propose a simple and natural randomized algorithm to embed a tree T in a given graph G. The algorithm can be viewed as a "self-avoiding tree-indexed random walk". The order of the tree T can be as large as a constant fraction of the order of the graph G, and the maximum degree of T can be close to the minimum degree of G. We show that our algorithm works in a variety of interesting settings. For example, we prove that any graph of minimum degree d without 4-cycles contains every tree of order d 2 and maximum degree at most d−2 d−2. As there exist d-regular graphs without 4-cycles and with O(d 2 ) vertices, this result is optimal up to constant factors. We prove similar nearly tight results for graphs of given girth and graphs with no complete bipartite subgraph Ks,t.

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Section: Discussionmentioning

confidence: 99%

Section: Extremal Questionsmentioning

confidence: 99%

A randomized embedding algorithm for trees

Sudakov¹,

Vondrák

2010

Combinatorica

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“…As remarked in [5], the condition that the average degree of the graph G is greater than k À 1 from the Erdo ÂÂs±So Âs conjecture is replaced in Loebl±Komlo Âs± So Âs conjecture by the condition that the medium degree of G is greater than k (for some special cases of the Erdo ÂÂs±So Âs conjecture see for example [8] as well as [2] and [7]). …”

Section: Conjecture 2 (Erdo ââS±so âS)mentioning

confidence: 98%

On the Loebl-Koml�s-S�s conjecture

Bazgan

Wo?niak³

2000

J. Graph Theory

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“…Their conjecture has attracted a fair amount of attention over the last decades. Partial solutions are given in , and in the early 1990s, Ajtai, Komlós, Simonovits, and Szemerédi announced a proof of this result for sufficiently large

m

. To see that the Erdős‐Sós conjecture is best possible, observe that no

(m - 1)

‐regular graph contains the star

K_{1, m}

as a subgraph.…”

Section: Introductionmentioning

confidence: 99%

A variant of the Erdős‐Sós conjecture

Havet

Reed

Stein

et al. 2019

Journal of Graph Theory

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A well‐known conjecture of Erdős and Sós states that every graph with average degree exceeding m−1 contains every tree with m edges as a subgraph. We propose a variant of this conjecture, which states that every graph of maximum degree exceeding m and minimum degree at least ⌊2m/3⌋ contains every tree with m edges. As evidence for our conjecture we show (a) for every m there is a g(m) such that the weakening of the conjecture obtained by replacing the first m by g(m) holds, and (b) there is a γ>0 such that the weakening of the conjecture obtained by replacing ⌊2m/3⌋ by (1−γ)m holds.

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The Erdős-Sós conjecture for graphs of girth 5

Cited by 52 publications

References 1 publication

A randomized embedding algorithm for trees

A randomized embedding algorithm for trees

On the Loebl-Koml�s-S�s conjecture

A variant of the Erdős‐Sós conjecture

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