Partitions and Coverings of Trees by Bounded-Degree Subtrees

This paper addresses the following questions for a given tree $T$ and integer $d\geq2$: (1) What is the minimum number of degree-$d$ subtrees that partition $E(T)$? (2) What is the minimum number of degree-$d$ subtrees that cover $E(T)$? We answer the first question by providing an explicit formula for the minimum number of subtrees, and we describe a linear time algorithm that finds the corresponding partition. For the second question, we present a polynomial time algorithm that computes a minimum covering. We then establish a tight bound on the number of subtrees in coverings of trees with given maximum degree and pathwidth. Our results show that pathwidth is the right parameter to consider when studying coverings of trees by degree-3 subtrees. We briefly consider coverings of general graphs by connected subgraphs of bounded degree

Extremal results in hypergraph theory via the absorption method

The so-called "absorbing method" was first introduced in a systematic way by Rödl, Ruciński and Szemerédi in 2006, and has found many uses ever since. Speaking in a general sense, it is useful for finding spanning substructures of combinatorial structures. We establish various results of different natures, in both graph and hypergraph theory, most of them using the absorbing method: 1. We prove an asymptotically best-possible bound on the strong chromatic number with respect to the maximum degree of the graph. This establishes a weak version of a conjecture of Aharoni, Berger and Ziv. 2. We determine asymptotic minimum codegree thresholds which ensure the existence of tilings with tight cycles (of a given size) in uniform hypergraphs. Moreover, we prove results on coverings with tight cycles. 3. We show that every 2-coloured complete graph on the integers contains a monochromatic infinite path whose vertex set is sufficiently "dense" in the natural numbers. This improves results of Galvin and Erdős and of DeBiasio and McKenney