On Schelp's problem for three odd long cycles
Introduction
We use Rado's notation and write whenever for each colouring of the edges of G with r colours there exists , such that some copy of in G has all its edges coloured by the ith colour. The smallest number N such that is called the Ramsey number for graphs and denoted . Schelp observed that quite often for G with vertices provided only that the minimum degree of G is large enough. He proved such a result for a pair of paths [11] and stated an appropriate conjecture for the minimum density for which it is true for a pair of cycles (see [9] and [12]). The conjecture has been proved to ‘asymptotically’ hold by Benevides et al. [1] and Gyárfás and Sárközy [6] (see also White [13] for an ‘extremal’ version of this problem).
Schelp's problem for a triple of cycles or paths seems to be much harder to settle. Let us recall that the asymptotic value of the Ramsey number for three long odd cycles was first found by Łuczak [10]. Then, it was used by Kohayakawa, Simonovits, and Skokan [8] to show that for sufficiently large odd n we have , confirming an old conjecture of Bondy and Erdős [3]. Our goal is to show the following theorem.
Theorem 1 For every there exists such that for every odd and every graph G on vertices with each colouring of the edges of G with three colours leads to a monochromatic cycle of length n.
Let us observe that the above result is ‘asymptotically’ best possible. Indeed, for an even n the complete balanced bipartite graph on n vertices is -regular, so there exists a graph G with 4n vertices with minimum degree whose edges can be decomposed into three bipartite graphs.
In the paper we use approach introduced in [10] based on the observation that, due to Szemerédi's Regularity Lemma, the existence of a long monochromatic cycle follows from the fact that the ‘reduced graph’ contains a non-bipartite monochromatic component with a large matching (for a formal exposition of the method see [5]). This is also true for graphs with large minimum degree and one can mimic the argument from [1] to show that Theorem 1 is a direct consequence of the following result.
Lemma 2 For every there exists such that for every and every graph G on vertices with the following holds. For each colouring of the edges of G with three colours there exists a non-bipartite monochromatic component which contains a matching saturating at least vertices.
The idea of the proof of Lemma 2 is to reduce the three-colour problem to a two-colour problem, which, in turn, is solved by using a number of “one-colour” results, i.e. facts on the existence of matchings contained in one components in graphs. Thus, the structure of the paper goes as follows. We start with some results on components and matchings we shall use later on. Then we state Lemma 8, crucial for our argument, which reduces the problem to two colours, and show how it implies Lemma 2 and so also Theorem 1. The next section is devoted to a long and technical proof of Lemma 8. We conclude the paper with some open problems and remarks.
Section snippets
Components and matchings
Let us start with the following fact on the size of the largest monochromatic components in dense graphs whose edges are coloured with three colours.
Lemma 3 Let be a graph on 4k vertices with . Then each colouring of the edges of G with three colours leads to a monochromatic component of size at least 2k.
Proof Let be the largest monochromatic component of G, say, in the first colour, and let . Consider the largest monochromatic bipartite component in the subgraph
Reduction to two colours
The key ingredient of our argument is a somewhat technical result, which reduces our problem to two colours. In a way one can consider it as a 2-colour version of 1-colour Lemma 6 from the last section. For sake of simplifying the notation let denote the graph . Moreover, here and below we shall omit all floors and ceilings and always assume that constants are chosen in such a way that the numbers which denote the number of vertices and degrees of
Proof of Lemma 8
Let us recall that is a graph obtained from the complete graph on vertices from which we have removed edges lying in three disjoint subsets , , of sizes , , respectively. We also set , , and . In this chapter we always assume that and , which imply that and so . Moreover, we consider a graph which is b-dense in G for some . Note that from the definition of b-density, if G is b
Final remarks and open problems
As we have already mentioned Kohayakawa, Skokan and Simonovits [8] proved that there exists such that for every odd we have . Thus, one can expect that the following to hold.
Conjecture There exists such that for every odd n, , and every graph G on vertices with minimum degree larger than each 3-colouring of the edges of G leads to a monochromatic cycle of length n.
One can also consider the case of triples of even cycles (which is very similar to the case of
Acknowledgement
We wish to thank an anonymous referee for his/her valuable comments and suggestions.
References (13)
- et al.
The 3-colored Ramsey number of even cycles
J. Combin. Theory Ser. B
(2009) - et al.
Ramsey numbers for cycles in graphs
J. Combin. Theory Ser. B
(1973) - et al.
The Ramsey number for a triple of long cycles
J. Combin. Theory Ser. B
(2007) - et al.
The 3-colored Ramsey number of odd cycles
Electron. Notes Discrete Math.
(2005) - et al.
A new class of Ramsey-Turán problems
Discrete Math.
(2010) J. Combin. Theory Ser. B
(1999)
Cited by (5)
On the restricted size Ramsey number for a pair of cycles
2024, Discrete MathematicsLarge monochromatic components in 3-colored non-complete graphs
2020, Journal of Combinatorial Theory. Series ALong monochromatic even cycles in 3-edge-coloured graphs of large minimum degree
2022, Journal of Graph Theory
- 1
The first author partially supported by National Science Centre, Poland, grant 2017/27/B/ST1/00873.