We prove that for any two graphs G and H , the edges of G can be strongly separated by a collection of linearly many subdivisions of H and single edges. This confirms a conjecture of Botler and Naia.

We establish minimum and maximum degree bounds for digraphs that ensure the containment of oriented trees of smaller order. We prove that, asymptotically, every large digraph of minimum semidegree above 2k/3 having vertices of out-degree and in-degree above k contains each large balanced antidirected bounded-degree tree with k arcs. Also, we show that, asymptotically, every large digraph of minimum semidegree above 3k/5 having vertices of out-degree and in-degree above 2k contains each large balanced antidirected bounded-degree tree with k arcs. Our result is restricted to balanced antidirected trees of bounded degree but may hold for other oriented trees as well.

Abstract. Kőnig’s theorem says that the vertex cover number of every bipartite graph is at most its matching number (in fact they are equal since, trivially, the matching number is at most the vertex cover number). An equivalent formulation of Kőnig’s theorem is that in every 2-coloring of the edges of a graph [Formula: see text], the number of monochromatic components needed to cover the vertex set of [Formula: see text] is at most the independence number of [Formula: see text]. We prove the following strengthening of Kőnig’s theorem: In every 2-coloring of the edges of a graph [Formula: see text], the number of monochromatic subgraphs of bounded diameter needed to cover the vertex set of [Formula: see text] is at most the independence number of [Formula: see text].

Abstract. We show that if [Formula: see text] is an [Formula: see text]-vertex digraph with more than [Formula: see text] arcs that does not contain any of three forbidden digraphs, then [Formula: see text] contains every antidirected tree on [Formula: see text] arcs. The forbidden digraphs are those orientations of [Formula: see text] where each of the vertices in the class of size two has either out-degree 0 or in-degree 0. This proves a conjecture of Addario-Berry et al. for a broad class of digraphs, and generalizes a result for [Formula: see text]-free graphs by Balasubramanian and Dobson. We also show that every digraph [Formula: see text] on [Formula: see text] vertices with more than [Formula: see text] arcs contains every antidirected [Formula: see text]-arc caterpillar, thus solving the above conjecture for caterpillars. This generalizes a result of Perles.

Gathering training data is a key step of any supervised learning task, and it is both critical and expensive. Critical, because the quantity and quality of the training data has a high impact on the performance of the learned function. Expensive, because most practical cases rely on humans-in-the-loop to label the data. The process of determining the correct labels is much more expensive than comparing two items to see whether they belong to the same class. Thus motivated, we propose a setting for training data gathering where the human experts perform the comparatively cheap task of answering pairwise queries, and the computer groups the items into classes (which can be labeled cheaply at the very end of the process). Given the items, we consider two random models for the classes: one where the set partition they form is drawn uniformly, the other one where each item chooses its class independently following a fixed distribution. In the first model, we characterize the algorithms that minimize the average number of queries required to cluster the items and analyze their complexity. In the second model, we analyze a specific algorithm family, propose as a conjecture that they reach the minimum average number of queries and compare their performance to a random approach. We also propose solutions to handle errors or inconsistencies in the experts' answers.

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.

We conjecture that every graph of minimum degree at least $\frac k2$ and maximum degree at least 2 k contains all trees with k edges as subgraphs. We prove an approximate version of this conjecture for trees of bounded degree and dense host graphs. Our result relies on a general embedding tool for embedding trees into graphs of certain structure. This tool also has implications for the Erdös--Sós conjecture and the $\frac 23$-conjecture. We prove an approximate version of both conjectures for bounded degree trees and dense host graphs.

In a series of four papers we prove the following relaxation of the Loebl--Komlós--Sós 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 $(\frac{1}{2}+\alpha)n$ 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 Szemerédi 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.

This is the second of a series of four papers in which we prove the following relaxation of the Loebl--Komlós--Sós 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 $(\frac{1}{2}+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first paper of this 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 third and fourth papers, we refine the structure and use it for embedding the tree $T$.

This is the third of a series of four papers in which we prove the following relaxation of the Loebl--Komlós--Sós 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 fourth paper, the refined structure will be used for embedding the tree $T$.

This is the last of a series of four papers in which we prove the following relaxation of the Loebl--Komlós--Sós 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--Komlós--Sós conjecture contains one of ten specific configurations. In this paper we embed the tree $T$ in each of the ten configurations.

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.

It is well known that in finite graphs, large complete minors/topological minors can be forced by assuming a large average degree. Our aim is to extend this fact to infinite graphs. For this, we generalize the notion of the relative end degree, which had been previously introduced by the first author for locally finite graphs, and show that large minimum relative degree at the ends and large minimum degree at the vertices imply the existence of large complete (topological) minors in infinite graphs with countably many ends.

A connected graph G is called t-perfect if its stable set polytope is determined by the nonnegativity, edge, and odd-cycle inequalities. Moreover, G is called strongly t-perfect if this system is totally dual integral. It is an open problem whether t-perfection is equivalent to strong t-perfection. We prove the equivalence for the class of claw-free graphs.

Bounds on the minimum degree and on the number of vertices attaining 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.