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

Packing spanning graphs from separable families

Let $\mathcal G$ be a separable family of graphs. Then for all positive constants $\epsilon$ and $\Delta$ and for every sufficiently large integer $n$, every sequence $G_1,\dotsc,G_t\in\mathcal G$ of graphs of order $n$ and maximum degree at most $\Delta$ such that $e(G_1)+\dotsb+e(G_t) \leq (1-\epsilon)\binom{n}{2}$ packs into $K_n$. This improves results of B\"ottcher, Hladk\'y, Piguet, and Taraz when $\mathcal G$ is the class of trees and of Messuti, R\"odl, and Schacht in the case of a general separable family. The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of Gy\'arf\'as (1976) for the case that all trees have maximum degree at most $\Delta$. The proof uses the local resilience of random graphs and a special multi-stage packing procedure

Monochromatic cycle covers in random graphs

A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every coloring of the edges of $K_n$ with $r$ colors, there is a cover of its vertex set by at most $f(r) = O(r^2 \log r)$ vertex-disjoint monochromatic cycles. In particular, the minimum number of such covering cycles does not depend on the size of $K_n$ but only on the number of colors. We initiate the study of this phenomena in the case where $K_n$ is replaced by the random graph $\mathcal G(n,p)$. Given a fixed integer $r$ and $p =p(n) \ge n^{-1/r + \varepsilon}$, we show that with high probability the random graph $G \sim \mathcal G(n,p)$ has the property that for every $r$-coloring of the edges of $G$, there is a collection of $f'(r) = O(r^8 \log r)$ monochromatic cycles covering all the vertices of $G$. Our bound on $p$ is close to optimal in the following sense: if $p\ll (\log n/n)^{1/r}$, then with high probability there are colorings of $G\sim\mathcal G(n,p)$ such that the number of monochromatic cycles needed to cover all vertices of $G$ grows with $n$.Comment: 24 pages, 1 figure (minor changes, added figure