Alexey Pokrovskiy

Partitioning edge-coloured complete graphs into monochromatic cycles and paths

Abstract A conjecture of Erdős, Gyarfas, and Pyber says that in any edge-colouring of a complete graph with r colours, it is possible to cover all the vertices with r vertex-disjoint monochromatic cycles. So far, this conjecture has been proven only for r = 2 . In this paper we show that in fact this conjecture is false for all r ⩾ 3 . In contrast to this, we show that in any edge-colouring of a complete graph with three colours, it is possible to cover all the vertices with three vertex-disjoint monochromatic paths , proving a particular case of a conjecture due to Gyarfas. As an intermediate result we show that in any edge-colouring of the complete graph with the colours red and blue, it is possible to cover all the vertices with a red path, and a disjoint blue balanced complete bipartite graph.

A linear bound on the Manickam-Miklós-Singhi conjecture

Suppose that we have a set S of n real numbers which have nonnegative sum. How few subsets of S of order k can have nonnegative sum? Manickam, Miklos, and Singhi conjectured that for n ? 4 k the answer is ( n - 1 k - 1 ) . This conjecture is known to hold when n is large compared to k. The best known bounds are due to Alon, Huang, and Sudakov who proved the conjecture when n ? 33 k 2 . In this paper we improve this bound by showing that there is a constant C such that the conjecture holds when n ? C k . This establishes the conjecture in a range which is a constant factor away from the conjectured bound. Progress is made 20 year old conjecture with a very simple statement.A new method is developed and used to improve known bounds on this conjecture.The conjecture is proved in a range which is a constant factor away from the conjectured range.

Highly linked tournaments

A (possibly directed) graph is k-linked if for any two disjoint sets of vertices { x 1 , ? , x k } and { y 1 , ? , y k } there are vertex disjoint paths P 1 , ? , P k such that P i goes from x i to y i . A theorem of Bollobas and Thomason says that every 22k-connected (undirected) graph is k-linked. It is desirable to obtain analogues for directed graphs as well. Although Thomassen showed that the Bollobas-Thomason Theorem does not hold for general directed graphs, he proved an analogue of the theorem for tournaments-there is a function f ( k ) such that every strongly f ( k ) -connected tournament is k-linked. The bound on f ( k ) was reduced to O ( k log ? k ) by Kuhn, Lapinskas, Osthus, and Patel, who also conjectured that a linear bound should hold. We prove this conjecture, by showing that every strongly 452k-connected tournament is k-linked.

Partitioning 3-coloured complete graphs into three monochromatic paths

In this paper we show that in any edge-colouring of the complete graph by three colours, it is possible to cover all the vertices by three disjoint monochromatic paths. This solves a particular case of a conjecture of Gyarfas. As an intermediate result, we show that in any edge colouring of the complete graph by two colours, it is possible to cover all the vertices by a monochromatic path and a disjoint monochromatic balanced complete bipartite graph.

Calculating Ramsey Numbers by Partitioning Colored Graphs

In this paper we prove a new result about partitioning coloured complete graphs and use it to determine certain Ramsey numbers exactly. The partitioning theorem we prove is that for k at least 1, in every edge colouring of a complete graph with the colours red and blue, it is possible to cover all the vertices with k disjoint red paths and a disjoint blue balanced complete (k+1)-partite graph. When the colouring is connected in red, we prove a stronger result - that it is possible to cover all the vertices with k red paths and a blue balanced complete (k+2)-partite graph. Using these results we determine the Ramsey number of a path on n vertices, versus a balanced complete k-partite graph, with m vertices in each part, whenever m-1 is divisible by n-1. This generalizes a result of Erdos who proved the m=1 case of this result. We also determine the Ramsey number of a path on n vertices versus the power of a path on n vertices. This solves a conjecture of Allen, Brightwell, and Skokan.

Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph Kn has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost n. In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured Kn has a rainbow cycle of length n − O(n3/4). One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edge-coloured Kn formed by the edges of a random set of colours has a similar edge distribution as a truly random graph with the same edge density. In particular, it has very good expansion properties.

Complex networks based on discrete-mode lasers

We propose a method to combine a number of discrete-mode lasers to construct a complex network with a nontrivial topology. The topological properties of this network are only encoded in the selection of the active modes for each laser, while the optical fibre coupling between the lasers is trivial. In a very simplified model of this network, we establish the stability of the fixed point with all lasing modes switched on.

Ramsey goodness of paths

Given a pair of graphs

A family of extremal hypergraphs for Ryser's conjecture

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On sets not belonging to algebras and rainbow matchings in graphs

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