Ramsey goodness of cycles

Given a pair of graphs G and H, the Ramsey number R(G, H) is the smallest N such that every red-blue coloring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If a graph G is connected, it is well known and easy to show that R(G, H) \geq (| G| - 1)(\chi (H) - 1) + \sigma (H), where \chi (H) is the chromatic number of H and \sigma (H) is the size of the smallest color class in a \chi (H)-coloring of H. A graph G is called H-good if R(G, H) = (| G| - 1)(\chi (H) - 1) + \sigma (H). The notion of Ramsey goodness was introduced by Burr and Erd\H os in 1983 and has been extensively studied since then. In this paper we show that if n \geq 1060| H| and \sigma (H) \geq \chi (H) 22, then the n-vertex cycle Cn is H-good. For graphs H with high \chi (H) and \sigma (H), this proves in a strong form a conjecture of Allen, Brightwell, and Skokan

The oriented size Ramsey number of directed paths

An oriented graph is a directed graph with no bi-directed edges, i.e. if xy is an edge then yx is not an edge. The oriented size Ramsey number of an oriented graph H, denoted by \vec{r}(H), is the minimum m for which there exists an oriented graph G with m edges, such that every 2-colouring of G contains a monochromatic copy of H. In this paper we prove that the oriented size Ramsey number of the directed paths on n vertices satisfies \vec{r}(\vec{P}_{n}) = \Omega (n^{2} log n). This improves a lower bound by Ben-Eliezer, Krivelevich and Sudakov. It also matches an upper bound by Bucić and the authors, thus establishing an asymptotically tight bound on \vec{r}(\vec{P}_{n}). We also discuss how our methods can be used to improve the best known lower bound of the k-colour version of \vec{r}(\vec{P}_{n})

Linearly many rainbow trees in properly edge-coloured complete graphs

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. The study of rainbow decompositions has a long history, going back to the work of Euler on Latin squares. In this paper we discuss three problems about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine’s Conjecture, and the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of Kn there are 10−6n edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the best known bounds on all these conjectures. Using our method we also show that every properly (n − 1)-edge-coloured Kn has n/9 − 6 edge-disjoint rainbow trees, giving further improvement on the Brualdi-Hollingsworth Conjectur

Ramsey goodness of paths

Given a pair of graphs G and H, the Ramsey number R(G, H) is the smallest N such that every red-blue coloring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If graph G is connected, it is well known and easy to show that R(G, H) ≥ (|G|−1)(χ(H)−1)+σ(H), where χ(H) is the chromatic number of H and σ the size of the smallest color class in a χ(H)- coloring of H. A graph G is called H-good if R(G, H) = (|G| − 1)(χ(H) − 1) + σ(H). The notion of Ramsey goodness was introduced by Burr and Erd˝os in 1983 and has been extensively studied since then. In this short note we prove that n-vertex path Pn is H-good for all n ≥ 4|H|. This proves in a strong form a conjecture of Allen, Brightwell, and Skokan

Edge-disjoint rainbow trees in properly coloured complete graphs

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. The study of rainbow decompositions has a long history, going back to the work of Euler on Latin squares. We discuss three problems about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine’s Conjecture, and the Kaneko-Kano-Suzuki Conjecture. The main result which we discuss is that in every proper edge-colouring of Kn there are 10−6n edge-disjoint isomorphic spanning rainbow trees. This simultaneously improves the best known bounds on all these conjectures. Using our method it is also possible to show that every properly (n−1)-edge-coloured Kn has n/9 edge-disjoint spanning rainbow trees, giving a further improvement on the Brualdi-Hollingsworth Conjectur

A COUNTEREXAMPLE TO STEIN'S EQUI-n-SQUARE CONJECTURE

In 1975 Stein conjectured that in every n × n array filled with the numbers 1, . . . , n with every number occuring exactly n times, there is a partial transversal of size n−1. In this note we show that this conjecture is false by constructing such arrays without partial transverals of size n − 1/ 42 ln n

Linearly many rainbow trees in properly edge-coloured complete graphs

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. The study of rainbow decompositions has a long history, going back to the work of Euler on Latin squares. In this paper we discuss three problems about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine’s Conjecture, and the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of Kn there are 10−6n edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the best known bounds on all these conjectures. Using our method we also show that every properly (n − 1)-edge-coloured Kn has n/9 − 6 edge-disjoint rainbow trees, giving further improvement on the Brualdi-Hollingsworth Conjecture

Ramsey Goodness of Bounded Degree Trees

Given a pair of graphs G and H, the Ramsey number R(G, H) is the smallest N such that every red-blue coloring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If a graph G is connected, it is well known and easy to show that R(G, H) ≥ (|G| − 1)(χ(H) − 1) + σ(H), where χ(H) is the chromatic number of H and σ(H) is the size of the smallest color class in a χ(H)-coloring of H. A graph G is called H-good if R(G, H) = (|G| − 1)(χ(H) − 1) + σ(H). The notion of Ramsey goodness was introduced by Burr and Erd˝os in 1983 and has been extensively studied since then. In this paper we show that if n ≥ Ω(|H| log4 |H|) then every n-vertex bounded degree tree T is H-good. The dependency between n and |H| is tight up to log factors. This substantially improves a result of Erd˝os, Faudree, Rousseau, and Schelp from 1985, who proved that n-vertex bounded degree trees are H-good when n ≥ Ω(|H| 4 ). MSC: 05C05, 05C5

Isomorphic bisections of cubic graphs

Graph partitioning, or the dividing of a graph into two or more parts based on certain conditions, arises naturally throughout discrete mathematics, and problems of this kind have been studied extensively. In the 1990s, Ando conjectured that the vertices of every cubic graph can be partitioned into two parts that induce isomorphic subgraphs. Using probabilistic methods together with delicate recolouring arguments, we prove Ando's conjecture for large connected graphs