Loebl-Komlos-Sos Conjecture: dense case

TL;DR

This paper proves a dense case of the Loebl-Komlos-Sos Conjecture, showing that large dense graphs contain all trees of a certain size as subgraphs, under specified degree conditions.

Contribution

It establishes the conjecture for dense graphs, extending previous results to a broader class of graphs with high minimum degree.

Findings

Dense graphs with high minimum degree contain all trees of a given size as subgraphs.

The result applies to graphs with at least half of the vertices having degree above a threshold.

The theorem holds for sufficiently large graphs, depending on the parameters.

Abstract

We prove a version of the Loebl-Komlos-Sos Conjecture for dense graphs. For each q>0 there exists a number $n_0\in \mathbb{N}$ such that for any n>n_0 and k>qn the following holds: if G be a graph of order n with at least n/2 vertices of degree at least k, then any tree of order k+1 is a subgraph of G.