4,154 research outputs found

    Monochromatic cycle covers in random graphs

    Full text link
    A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every coloring of the edges of KnK_n with rr colors, there is a cover of its vertex set by at most f(r)=O(r2logr)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 KnK_n but only on the number of colors. We initiate the study of this phenomena in the case where KnK_n is replaced by the random graph G(n,p)\mathcal G(n,p). Given a fixed integer rr and p=p(n)n1/r+εp =p(n) \ge n^{-1/r + \varepsilon}, we show that with high probability the random graph GG(n,p)G \sim \mathcal G(n,p) has the property that for every rr-coloring of the edges of GG, there is a collection of f(r)=O(r8logr)f'(r) = O(r^8 \log r) monochromatic cycles covering all the vertices of GG. Our bound on pp is close to optimal in the following sense: if p(logn/n)1/rp\ll (\log n/n)^{1/r}, then with high probability there are colorings of GG(n,p)G\sim\mathcal G(n,p) such that the number of monochromatic cycles needed to cover all vertices of GG grows with nn.Comment: 24 pages, 1 figure (minor changes, added figure

    Minimum degree conditions for monochromatic cycle partitioning

    Get PDF
    A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any rr-edge-coloured complete graph has a partition into O(r2logr)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 cc such that any rr-edge-coloured graph on nn vertices with minimum degree at least n/2+crlognn/2 + c \cdot r \log n has a partition into O(r2)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

    Covering graphs by monochromatic trees and Helly-type results for hypergraphs

    Full text link
    How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given rr-edge-coloured graph GG? These problems were introduced in the 1960s and were intensively studied by various researchers over the last 50 years. In this paper, we establish a connection between this problem and the following natural Helly-type question in hypergraphs. Roughly speaking, this question asks for the maximum number of vertices needed to cover all the edges of a hypergraph HH if it is known that any collection of a few edges of HH has a small cover. We obtain quite accurate bounds for the hypergraph problem and use them to give some unexpected answers to several questions about covering graphs by monochromatic trees raised and studied by Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Gir\~ao, Letzter and Sahasrabudhe.Comment: 20 pages including references plus 2 pages of an Appendi

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

    Get PDF
    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

    Full text link
    In 1995, Erd\H{o}s and Gy\'arf\'as proved that in every 22-colouring of the edges of KnK_n, there is a vertex cover by 2n2\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,sr,s, what is the smallest number pcr,s(Kn)\text{pc}_{r,s}(K_n) such that in every colouring of the edges of KnK_n with rr colours, there exists a vertex cover of KnK_n by pcr,s(Kn)\text{pc}_{r,s}(K_n) monochromatic paths using altogether at most ss different colours? For fixed integers r>sr>s and as nn\to\infty, we prove that pcr,s(Kn)=Θ(n1/χ)\text{pc}_{r,s}(K_n) = \Theta(n^{1/\chi}), where χ=max{1,2+2sr}\chi=\max{\{1,2+2s-r\}} is the chromatic number of the Kneser gr aph KG(r,rs)\text{KG}(r,r-s). More generally, if one replaces KnK_n by an arbitrary nn-vertex graph with fixed independence number α\alpha, then we have pcr,s(G)=O(n1/χ)\text{pc}_{r,s}(G) = O(n^{1/\chi}), where this time around χ\chi is the chromatic number of the Kneser hypergraph KG(α+1)(r,rs)\text{KG}^{(\alpha+1)}(r,r-s). This result is tight in the sense that there exist graphs with independence number α\alpha for which pcr,s(G)=Ω(n1/χ)\text{pc}_{r,s}(G) = \Omega(n^{1/\chi}). This is in sharp contrast to the case r=sr=s, where it follows from a result of S\'ark\"ozy (2012) that pcr,r(G)\text{pc}_{r,r}(G) depends only on rr and α\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 dd-regular graphs

    Partitioning random graphs into monochromatic components

    Full text link
    Erd\H{o}s, Gy\'arf\'as, and Pyber (1991) conjectured that every rr-colored complete graph can be partitioned into at most r1r-1 monochromatic components; this is a strengthening of a conjecture of Lov\'asz (1975) in which the components are only required to form a cover. An important partial result of Haxell and Kohayakawa (1995) shows that a partition into rr monochromatic components is possible for sufficiently large rr-colored complete graphs. We start by extending Haxell and Kohayakawa's result to graphs with large minimum degree, then we provide some partial analogs of their result for random graphs. In particular, we show that if p(27lognn)1/3p\ge \left(\frac{27\log n}{n}\right)^{1/3}, then a.a.s. in every 22-coloring of G(n,p)G(n,p) there exists a partition into two monochromatic components, and for r2r\geq 2 if p(rlognn)1/rp\ll \left(\frac{r\log n}{n}\right)^{1/r}, then a.a.s. there exists an rr-coloring of G(n,p)G(n,p) such that there does not exist a cover with a bounded number of components. Finally, we consider a random graph version of a classic result of Gy\'arf\'as (1977) about large monochromatic components in rr-colored complete graphs. We show that if p=ω(1)np=\frac{\omega(1)}{n}, then a.a.s. in every rr-coloring of G(n,p)G(n,p) there exists a monochromatic component of order at least (1o(1))nr1(1-o(1))\frac{n}{r-1}.Comment: 27 pages, 2 figures. Appears in Electronic Journal of Combinatorics Volume 24, Issue 1 (2017) Paper #P1.1
    corecore