Name
Abstract | ||
|---|---|---|
In a series of four papers we prove the following relaxation of the Loebl-Komlos-Sos conjecture: For every alpha > 0 there exists a number k(0) such that for every k > k(0), every n-vertex graph G with at least (1/2 + alpha) n vertices of degree at least (1 + alpha)k contains each tree T of order k as a subgraph. The method to prove our result follows a strategy similar to approaches that employ the Szemeredi regularity lemma: We decompose the graph G, find a suitable combinatorial structure inside the decomposition, and then embed the tree T into G using this structure. Since for sparse graphs G, the decomposition given by the regularity lemma is not helpful, we use a more general decomposition technique. We show that each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. In this paper, we introduce this novel decomposition technique. In the three follow-up papers, we find a suitable combinatorial structure inside the decomposition, which we then use for embedding the tree. |
Year | DOI | Venue |
|---|---|---|
2017 | 10.1137/140982842 | SIAM JOURNAL ON DISCRETE MATHEMATICS |
Keywords | DocType | Volume |
extremal graph theory,Loebl-Komlos-Sos conjecture,tree embedding,regularity lemma,sparse graph,graph decomposition | Journal | 31 |
Issue | ISSN | Citations |
2 | 0895-4801 | 0 |
PageRank | References | Authors |
0.34 | 0 | 6 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Jan Hladký | 1 | 113 | 18.59 |
| János Komlós | 2 | 1254 | 255.25 |
| Diana Piguet | 3 | 12 | 7.65 |
| Miklós Simonovits | 4 | 578 | 98.20 |
| maya stein | 5 | 81 | 15.65 |
| Endre Szemerédi | 6 | 2102 | 363.27 |