296 research outputs found
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
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