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We propose the following conjecture: For every fixed $\alpha\in [0,\frac 13)$, each graph of minimum degree at least $(1+\alpha)\frac k2$ and maximum degree at least $2(1-\alpha)k$ contains each tree with $k$ edges as a subgraph. Our main result is an approximate version of the conjecture for bounded degree trees and large dense host graphs. We also show that our conjecture is asymptotically best possible. The proof of the approximate result relies on a second result, which we believe to be interesting on its own. Namely, we can embed any bounded degree tree into host graphs of minimum/maximum degree asymptotically exceeding $\frac k2$ and $\frac 43k$, respectively, as long as the host graph avoids a specific structure.

This is the last paper 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 two papers of this series, we decomposed the host graph $G$, and found a suitable combinatorial structure inside the decomposition. In the third paper, we refined this structure, and proved that any graph satisfying the conditions of the above approximate version of the Loebl-Komlos-Sos Conjecture contains one of ten specific configurations. In this paper we embed the tree $T$ in each of the ten configurations., Comment: 81 pages, 12 figures. A fix reflecting the change of Preconfiguration Clubs in Paper III, additional small changes

We show that for any coloring of the edges of the complete bipartite graph $K_{n,n}$ with three colors there are five disjoint monochromatic cycles which together cover all but $o(n)$ of the vertices. In the same situation, 18 disjoint monochromatic cycles together cover all vertices.

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 changed

This is the second 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; this decomposition might be viewed as an analogue of a regular partition for sparse graphs. In the present paper, we find a combinatorial structure inside this decomposition. In the last two papers, we refine the structure and use it for embedding the tree $T$., Comment: 38 pages, 4 figures; new is Section 5.1.1; accepted to SIDMA

Bounds on the minimum degree and on the number of vertices at- taining it have been much studied for finite edge-/vertex-minimally k- connected/k-edge-connected graphs. We give an overview of the results known for finite graphs, and show that most of these carry over to infinite graphs if we consider ends of small degree as well as vertices., Comment: 15 pages