Local colourings and monochromatic partitions in complete bipartite graphs

We show that for any $2$-local colouring of the edges of the balanced complete bipartite graph $K_{n,n}$, its vertices can be covered with at most~$3$ disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite $r$-locally coloured graph with $O(r^2)$ disjoint monochromatic cycles.\\ We also determine the $2$-local bipartite Ramsey number of a path almost exactly: Every $2$-local colouring of the edges of $K_{n,n}$ contains a monochromatic path on $n$ vertices.Comment: 18 page

Minimum degree conditions for monochromatic cycle partitioning

A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any $r$-edge-coloured complete graph has a partition into $O(r^2 \log r)$ monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, we show that there exists a constant $c$ such that any $r$-edge-coloured graph on $n$ vertices with minimum degree at least $n/2 + c \cdot r \log n$ has a partition into $O(r^2)$ monochromatic cycles. We also provide constructions showing that the minimum degree condition and the number of cycles are essentially tight.Comment: 22 pages (26 including appendix

Partitioning edge-coloured complete graphs into monochromatic cycles and paths

A conjecture of Erd\H{o}s, Gy\'arf\'as, 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 > 2. 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 Gy\'arf\'as. 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.Comment: 25 pages, 3 figure

Vertex covers by monochromatic pieces - A survey of results and problems

This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects. It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014.Comment: Discrete Mathematics, 201

Vertex covering with monochromatic pieces of few colours

In 1995, Erd\H{o}s and Gy\'arf\'as proved that in every $2$-colouring of the edges of $K_n$, there is a vertex cover by $2\sqrt{n}$ monochromatic paths of the same colour, which is optimal up to a constant factor. The main goal of this paper is to study the natural multi-colour generalization of this problem: given two positive integers $r,s$, what is the smallest number $\text{pc}_{r,s}(K_n)$ such that in every colouring of the edges of $K_n$ with $r$ colours, there exists a vertex cover of $K_n$ by $\text{pc}_{r,s}(K_n)$ monochromatic paths using altogether at most $s$ different colours? For fixed integers $r>s$ and as $n\to\infty$, we prove that $\text{pc}_{r,s}(K_n) = \Theta(n^{1/\chi})$, where $\chi=\max{\{1,2+2s-r\}}$ is the chromatic number of the Kneser gr aph $\text{KG}(r,r-s)$. More generally, if one replaces $K_n$ by an arbitrary $n$-vertex graph with fixed independence number $\alpha$, then we have $\text{pc}_{r,s}(G) = O(n^{1/\chi})$, where this time around $\chi$ is the chromatic number of the Kneser hypergraph $\text{KG}^{(\alpha+1)}(r,r-s)$. This result is tight in the sense that there exist graphs with independence number $\alpha$ for which $\text{pc}_{r,s}(G) = \Omega(n^{1/\chi})$. This is in sharp contrast to the case $r=s$, where it follows from a result of S\'ark\"ozy (2012) that $\text{pc}_{r,r}(G)$ depends only on $r$ and $\alpha$, but not on the number of vertices. We obtain similar results for the situation where instead of using paths, one wants to cover a graph with bounded independence number by monochromatic cycles, or a complete graph by monochromatic $d$-regular graphs

Monochromatic cycle partitions in local edge colourings

An edge colouring of a graph is said to be an $r$-local colouring if the edges incident to any vertex are coloured with at most $r$ colours. Generalising a result of Bessy and Thomass\'e, we prove that the vertex set of any $2$-locally coloured complete graph may be partitioned into two disjoint monochromatic cycles of different colours. Moreover, for any natural number $r$, we show that the vertex set of any $r$-locally coloured complete graph may be partitioned into $O(r^2 \log r)$ disjoint monochromatic cycles. This generalises a result of Erd\H{o}s, Gy\'arf\'as and Pyber.Comment: 10 page