Partitioning 3-colored complete graphs into three monochromatic cycles

We show in this paper that in every 3-coloring of the edges of Kn all but o(n) of its vertices can be partitioned into three monochromatic cycles. From this, using our earlier results, actually it follows that we can partition all the vertices into at most 17 monochromatic cycles, improving the best known bounds. If the colors of the three monochromatic cycles must be different then one can cover ( 3 4 − o(1))n vertices and this is close to best possible

The Approximate Loebl-Koml\'os-S\'os Conjecture III: The finer structure of LKS graphs

This is the third of a series of four papers in which 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 $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first paper of the series, we gave a decomposition of the graph $G$ into several parts of different characteristics. In the second paper, we found a combinatorial structure inside the decomposition. In this paper, we will give a refinement of this structure. In the forthcoming fourth paper, the refined structure will be used for embedding the tree $T$.Comment: 59 pages, 4 figures; further comments by a referee incorporated; this includes a subtle but important fix to Lemma 5.1; as a consequence, Preconfiguration Clubs was change