Proof of the ( n / 2 − n / 2 − n / 2 ) Conjecture for large
نویسنده
چکیده
A conjecture of Loebl, also known as the (n/2 − n/2 − n/2) Conjecture, states that if G is an n-vertex graph in which at least n/2 of the vertices have degree at least n/2, then G contains all trees with at most n/2 edges as subgraphs. Applying the Regularity Lemma, Ajtai, Komlós and Szemerédi [Graph theory, combinatorics, and algorithms, Vol. 2, 1135–1146, 1995] proved an approximate version of this conjecture. We prove it exactly for sufficiently large n. This immediately gives a tight upper bound for the Ramsey number of trees, and partially answers a conjecture of Burr and Erdős.
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تاریخ انتشار 2004