Monochromatic subgraphs of 2-edge-colored graphs

Transcription

1 Monochromatic subgraphs of 2-edge-colored graphs Luke Nelsen, Miami University June 10, 2014 Abstract Lehel conjectured that for all n, any 2-edge-coloring of K n admits a partition of the vertex set into a red cycle and a blue cycle. This conjecture led to a significant amount of work on related questions and was eventually proven for all n by Bessy and Thomassé. Balogh, Barát, Gerbner, Gyárfás, and Sárközy conjectured a stronger statement for large n: that if δ(g) > 3n/4, then any 2-edge-coloring of G admits such a partition. Balogh, et al. use regularity and blow-up techniques to cover all but γn vertices if δ(g) > ( γ)n. DeBiasio and the author use the absorbing method to prove their conjecture. This paper provides an overview of the history of the problem and the proof techniques of Balogh, et al. and DeBiasio and Nelsen. 1 The History behind Covering 2-edge-colored Graphs 1.1 Lehel s Conjecture and Generalizations Since Ramsey-type problems ask for which n any 2-edge-colored K n must admit certain monochromatic subgraphs, a problem investigating a partition of 2-edge-colored 1 complete graphs into certain monochromatic subgraphs is naturally related to Ramsey-type problems. In proving that R(P l, P l ) = 3l 2, Gerencsér and Gyárfás observed that the vertex set of any 2-colored Kn can be partitioned into a monochromatic cycle and a monochromatic path of the other color 2 [9]. A natural question to follow this observation is, can the vertex set of every 2-colored K n be partitioned into a red cycle and a blue cycle? Conjecture 1.1. For all n, any 2-coloring of K n admits a partition of the vertex set by a red cycle and a blue cycle. 3 Conjecture 1.1 is attributed to Lehel ([2]) but was first accessibly published in a paper by Gyárfás ([13]). Perhaps the most natural generalization of Statement 1.1 is the following question: Question 1.2. Does every r-coloring of K n admit a partition of the vertex set into r cycles of different colors? 1 Since the investigation of this paper never discusses vertex colorings of graphs, we will from this point forward refer to edge-colorings as colorings. 2-colorings are {red, blue}-edge-colorings. More generally, r-colorings are {1, 2,..., r}- colorings. 2 Clearly, if every edge of K n is colored red, then a non-degenerate blue cycle or path does not exist. We use the convention that the empty set, a single vertex K 2 are monochromatic cycles and paths. 3 Since this statement was conjectured by Lehel, we refer to such a partition as a Lehel partition. So Statement 1.1 may read, For all n, any 2-coloring of K n admits a Lehel partition. 1

2 Question 1.2 for r = 2 is Lehel s Conjecture. However, for r 3, Question 1.2 is false. Consider the following 3-coloring of K n, n 8: Partition V (K n ) into balanced parts V 1, V 2, V 3, V 4. Color [V 1, V 2 ] and [V 3, V 4 ] with color 1. Color [V 1, V 3 ] and [V 2, V 4 ] with color 2. Color [V 1, V 4 ] and [V 2, V 3 ] with color 3. Color E(V i ) with color 1 for i = 1, 2, 3, 4. Figure 1 This 3-coloring does not admit three monochromatic cycles of different colors, and this counterexample can be generalized for r 3. So for r 3, we remove from Question 1.2 the restriction that the r colors must be different. A 1989 paper by Gyárfás ([14]) and a 1991 paper by Erdős, Gyárfás and Pyber ([7]) brought further investigation into several such questions. Let c(r), p(r), t(r) be the minimum number of disjoint monochromatic cycles, paths, trees (respectively) required to partition the vertex set for any r-coloring of K n for all n. 4 Then we have the following conjectures: Conjecture 1.3 (Erdős, Gyárfás, Pyber 1991). c(r) = r. Conjecture 1.4 (Gyárfás 1989). p(r) = r. Conjecture 1.5 (Erdős, Gyárfás and Pyber 1991). t(r) = r Results of Questions from Section 1.1 Lehel s Conjecture did not see results until after the development of the Blow-Up Lemma; Luczak, Rödl and Szemerédi used it in [19] to prove Lehel s Conjecture for extremely large n (1998). Allen later proved Lehel s Conjecture for sufficiently large n using Ramsey-related techniques [1] (2008), and later Bessy and Thomassé gave an elementary proof for all n [5] (2010). In the meantime, Erdős, Gyárfás and Pyber [7] proved c(r) cr 2 log r (1991). This bound was improved to c(r) cr log r in [10] (2006), and Bessy and Thomassé s proof of Lehel s Conjecture in 2010 proves c(r) = 2. More specific results were obtained for r = 3 in 2011: c(3) 17, and also 4 It is not immediately trivial that c(r), p(r), t(r) are functions of r, but it is worth observing that t(r) p(r) c(r). 2

3 all but o(n) vertices can be covered by three monochromatic cycles [11]. However, Pokrovskiy [21] reently showed by construction that for each r 3, there are infinitely many counterexamples to Conjecture 1.3. So c(r) r + 1 except for r = 2; we will not obtain Conjecture 1.4 via a proof of Conjecture 1.3. However, there are some results for p(r) and t(r). By the observation in [9] mentioned previously, p(2) = 2. Pokrovskiy [21] proved p(3) = 3. Erdős, Gyárfás and Pyber in [7] show that t(r) r 1 and that t(3) = 2. (Note that t(2) = 1 since for any graph G, G or G is connected.) In 1996, Haxell and Kohayakawa proved that t(r) r for n n 0 (r) [15]. We now move to another, more recent kind of question that laid the framework for the author s research. 1.3 Moving from Complete Graphs to δ(g) 3n/4 As previously mentioned, in [9] Gerencsér and Gyárfás proved the following: Theorem 1.6 (Gerencsér and Gyárfás 1967). For any 2-coloring of K n, there is a monochromatic P k such that k 2n 3. Faudree, Lesniak and Schiermeyer proved a stronger result about the circumference of the monochromatic graphs in [8]: Theorem 1.7 (Faudree, Lesniak and Schiermeyer 2009). For any 2-coloring of K n where n 6, there is a monochromatic C k such that k 2n 3. So proving a stronger result about monochromatic cycles resulted in a result for monochromatic paths. In [23], Schelp formulated a conjecture similar to Theorem 1.6 that weakened the minimum degree condition of the graph being 2-colored: Conjecture 1.8 (Schelp 2012). For n large enough, if G is a graph on n vertices with δ(g) > 3n 4, then any 2-coloring of G contains a monochromatic P 2n 3. Schelp collected and presented questions from this type of problem, which he poses as follows: Question 1.9 (Schelp 2012). For which graphs G does there exist a constant 0 < c < 1 such that when H is a graph of order the Ramsey number r(g) with δ(h) > c H, then any 2-edge coloring of H contains a monochromatic copy of G? So Conjecture 1.8 is a type of Question 1.9. Another question of this type had been posed by Li, Nikiforov and Schelp in [18]: Question 1.10 (Li, Nikiforov, Schelp 2010). If 0 < c < 1 and G is a 2-colored graph of sufficiently large order n with δ(g) > cn, how long are the monochromatic cycles? Benevides, Luczak, Scott, Skokan and White prove the following, which proves Conjecture 1.8 asymptotically: Theorem 1.11 (Benevides, et al. 2012). For δ > 0 and G of order n n 0 (δ) with δ(g) 3n 4, any 2-coloring of G admits a monochromatic cycle of length at least ( 2 3 δ)n. 3

4 Benevides, et al. conjecture that the answer to Question 1.10 for c = 3 4 is 2n 3. Now, how would one adapt Lehel s Conjecture in the spirit of Question 1.9? The question should sound like, Does there exist a constant 0 < c < 1 such that when H is a graph of order n with δ(h) > cn, then any 2-edge coloring of H admits a Lehel partition? And if so, what is c? In [3], a conjecture is supplied to meet the question and an approximate result is given: Conjecture 1.12 (Balogh, et al ). Let G be a graph on n vertices and δ(g) > 3n/4. Then any 2-coloring of G admits a Lehel partition. Theorem 1.13 (Balogh, et al ). For η > 0, let G be a graph on n n 0 (η) vertices and δ(g) > ( η)n. Then any 2-coloring of G admits two vertex-disjoint monochromatic cycles of different colors covering at least (1 η)n vertices of G. So if Conjecture 1.12 is true, then the result of Lehel s Conjecture holds for graphs with minimum degree greater than 3/4 order. DeBiasio and the author prove a stronger form of Theorem 1.13 in [6] to obtain an asymptotic version of Conjecture 1.12: Theorem 1.14 (DeBiasio and Nelsen 2014-). For η > 0, let G be a graph on n n 0 (η) vertices and δ(g) ( η)n. Then any 2-coloring of G admits a Lehel partition. The techniques used in the proof of Theorem 1.14 are the main topic of this paper. Section 7 discusses the proof of an exact result of Conjecture 1.12 for large n: Theorem 1.15 (DeBiasio and Nelsen 2014-). Let G be a graph of sufficiently large order n with δ(g) 3n 4. Then any 2-coloring of G admits a Lehel partition. 2 Some Comments about Conjecture Why c = 3 4? Upon first glance, it may not be obvious why the c value in Conjecture 1.8, Theorem 1.11 and Conjecture 1.12 is 3/4. The following construction shows that if c = 3 4, then this value is tight: Partition V (G) into A 1, A 2, A 3, A 4 with 1 A i A j 1 for all i, j [4]. Put no edges between A 1 and A 3 or between A 2 and A 4. Join A 1 completely to A 2 and join A 3 completely to A 4 ; color these edges red. Join A 1 completely to A 3 and join A 2 completely to A 4 ; color these edges blue. Color the edges in each A i arbitrarily. (See Figure 2.) The resulting graph G has minimum degree 3 G 4 of vertices that one red cycle and one blue cycle can cover (even permitting intersection) is 2.2 Two Counterexamples on n = 9 1, but the maximum number. In the author s work with Louis DeBiasio a counterexample of small order was encountered. (See Figure 3.) There exists no red cycle and blue cycle which partition the vertex set of this graph in the given 2-coloring. A note of interest is that this example is not particularly tight in two ways: 3 G 4 4

5 Figure 2 Figure 3 First, in that the minimum degree of the graph is 7 = 3n+1 4 ; so the failure of this 2-coloring is not merely because of equality in the minimum degree condition. Second, the coloring itself is not unique in its failure; if x 3 y 2 is colored either red or blue, the 2-coloring still fails to admit a Lehel partition. Thus Conjecture 1.12 does not hold for n = 9. It is likely, however, that Balogh, et al. intended their conjecture for large n. 3 An Overview of Relevant Tools The proof of Theorem 1.13 utilizes the regularity and blow-up lemmas of Szemerédi. The proof of Theorem 1.14 adapts the proof of Theorem 1.13 using a technique similar to that used by Rödl, Ruciński and Szemerédi in [22]. This section introduces the usefulness of the regularity and blow-up lemmas. 5 5 We borrow the formulations of the Regularity Lemma and the Key Lemma from the survey by Komlós and Simonovits ([17]). 5

6 3.1 The Regularity Lemma To discuss the Regularity Lemma, we first need a few definitions: Definition 3.1. For a graph G and disjoint X, Y V (G), the density of the pair (X, Y ) is d(x, Y ) = e(x, Y ) X Y, where e(x, Y ) is the number of edges with one endpoint in X and the other endpoint in Y. Definition 3.2. Let ε > 0. For a graph G and disjoint A, B V (G), the pair (A, B) is ε-regular if for all X A, Y B such that X > ε A and Y > ε B, then d(x, Y ) d(a, B) < ε. So if a bipartite graph is ε-regular for a small ε > 0, then there is a high level of uniformity in the distribution of the edges between the two parts; no subsets of decent size have a density that deviates substantially from that of the graph. This uniformity means that an ε-regular pair behaves much like one would expect a random bipartite graph to behave. This is a key point and is formalized primarily in two ways. First, at most a small proportion of the vertices can have degree that deviates from the density: Fact 1. Let (A, B) be an ε-regular pair. If Y B with Y > ε B, then the size of is at most ε A. 6 {x A : deg(x, Y ) (d(a, B) ε) Y } Second, vertices in one part often have the expected number of common neighbors in the other part: Fact 2. Let l 1 and let (A, B) be an ε-regular pair with density d = d(a, B). If Y B with Y > ε B, then the size of { ( l } (x 1, x 2,..., x l ) A : Y N(x i )) (d ε)l Y is at most ε A. So working with ε-regular pairs is in many ways as convenient as working with random bipartite graphs. Now we are ready for the Regularity Lemma: 7 Lemma 3.1 (Regularity Lemma (Degree Form)). For every ε > 0 there exists M = M(ε) such that if G is a graph and d [0, 1], then there is a partition of V (G) into V 0, V 1,..., V k and there is a subgraph G G such that the following are true: k M 6 There is a similar bound for vertices with degree exceeding (d(a, B) + ε) Y. 7 The form given for the Regularity Lemma is called the Degree Form and is in some ways easier to work with. For the proofs discussed in the following sections, we use a 2-colored form of the Regularity Lemma which yields a partition that yields the same results simultaneously on the red and blue graphs. i=1 6

7 V 0 ε V (G) each V i, i 1, are of size m ε V (G) deg G (v) > deg G (v) (d + ε) V (G) for all v V (G) e(g [V i ]) = 0 for all i 1 in G, all pairs (V i, V j ) (for 1 i < j k) are ε-regular with a density 0 or at least d. In general, the Regularity Lemma says that there is a partition into a bounded number of parts (where V 0 is a leftover set so that V 1,..., V k are of equal size), almost all of which form ε-regular pairs. The degree form says that there is a partition such that, upon deleting the edges within each V i and in the irregular pairs and in the pairs with density less than d, yields a subgraph G for which all pairs are ε-regular and for which each vertex loses only small degree. A pure graph G = G \ V 0 is obtained from this lemma which is almost all of the original graph G but is conveniently partitioned into independent sets of equal size, each pair of which is empty or ε-regular with degree at least d. This result is trivial for small V (G) but powerful for large V (G). 3.2 The Blow-up Lemma The Regularity Lemma serves our purposes in that as a corollary we are able to embed certain subgraphs into G. 8 This is made precise by the Key Lemma, and vastly improved upon by the Blow-up Lemma. First, a few more definitions: Definition 3.3. Given an application of the Regularity Lemma (Degree Form) to a pure graph G, the reduced graph R is defined by V (R) = {v 1,..., v k } where {v i, v j } E(R) iff (V i, V j ) is ε-regular with d(v i, V j ) d. An important observation about reduced graphs is that miminum degree is mostly preserved: Proposition 3.2. Let R be the reduced graph obtained by application of the Regularity Lemma to G with parameters ε and d. If δ(g) cn and 0 < 2ε d c/2, then δ(r) (c 2d) V (R). 9 Definition 3.4. Given a graph R and positive integer t, define R(t) to be the graph obtained from R by replacing each vertex v V (R) with an independent set V x of t vertices, and joining u V x to v V y iff {x, y} E(R). That is, by replacing the edges of R with complete bipartite K t,t s. 10 So in reducing G to R and then blowing up to R(m), we obtain a graph that is very similar in structure to G ; so similar that one would expect a subgraph (of bounded degree) of R(m) to be a subgraph also of G. By building vertex-by-vertex according to the natural bijection between the independent sets V i V vi, we should complete the building process so long as the subgraph does not greedily use too many vertices from a set V i which is guaranteed by a bounded degree. This is stated formally as the Key Lemma: Lemma 3.3 (Key Lemma). Given d > ε > 0, a graph R, and positive integer m, let G be a graph constructed by replacing every vertex of R by m vertices and replacing the edges of R with ε-regular pairs of density at least d. Let H be a subgraph of R(t) on h vertices and maximum degree > 0, 8 So it is safe to assume that d > 0. 9 From [16]. 10 Sometimes R(t) is called R blow-up of t. 7

8 and let δ = d ε and ε 0 = δ /(2 ). If ε ε 0 and t 1 ε 0 m, then H G. In fact, there are at least (ε 0 m) h labelled copies of H in G. The Key Lemma is useful in that it can take a potential subgraph restricted primarily only by maximum degree and leverage the power of ε-regular pairs to guarantee an embedding of the subgraph. But the Key Lemma does not leverage the full power of ε-regular pairs; given that the edges are distributed so uniformly inside the pairs, surely we assume much loss in using the greedy building algorithm. In this sense the Blow-up Lemma improves upon the Key Lemma. But first, another definition: Definition 3.5. For a graph G and disjoint A, B V (G), the pair (A, B) is (ε, δ)-super-regular if for all X A, Y B such that X > ε A and Y > ε B, then e(x, Y ) > δ X Y and furthermore, deg(a) > δ B for all a A and deg(b) > δ A for all b B. So an (ε, δ)-super-regular pair is an ε-regular pair with the added condition that there are zero vertices whose degree falls below a certain threshold; there are no ill-behaved vertices. Theorem 3.4 ((Blow-up Lemma) Komlós, Sárközy and Szemerédi 1997). Given a graph R of order r and positive parameters δ,, there exists a positive ε = ε(δ,, r) such that the following holds. Let n 1, n 2,..., n r be arbitrary positive integers and let us replace the vertices v 1, v 2,..., v r of R with pairwise disjoint sets V 1, V 2,..., V r of sizes n 1, n 2,.., n r (blowing up). We construct two graphs on the same vertex-set V = V i. The first graph R is obtained by replacing each edge {v i, v j } of R with the complete bipartite graph between the corresponding vertex-sets V i and V j. A sparser graph G is constructed by replacing each edge {v i, v j } arbitrarily with an (ε, δ)- super-regular pair between V i and V j. If a graph H with (H) < is embeddable into R, then it is already embeddable into G. The Blow-up Lemma makes use of the likelihood of success using a randomized algorithm (rather than assuming the worst possible result) and super-regular pairs to obtain much larger subgraphs; there is no longer the restriction t 1 ε 0 m from the Key Lemma. So the Blow-up Lemma is a powerful tool for obtaining spanning or nearly spanning subgraphs for large n. 4 Proof of Balogh, et al. A key aspect of the proof of Theorem 1.13 lies in the application of a structural lemma elementary results about the component structure of 2-colored graphs with minimum degree 3n 4 11 to obtain a result in a reduced graph. By applying blow-up, 12 we are able to use the result on the reduced graph to obtain nearly-spanning monochromatic cycles. Lemma 4.1 (Balogh, et al.). Let G with δ(g) 3n/4 be {1, 2}-colored and H i be the largest component in color i for i = 1, 2. Then one of the following holds: (i) H 1 or H 2 spans V (G) (ii) H i 3n/4 for i = 1, 2 and V (G) = V (H 1 ) V (H 2 ). Furthermore, H 1 H 2 contains a perfect matching. 11 Here we use n as the order of a graph understood. 12 To apply regularity/blow-up, we assume that G is large. 8

9 Now the proof of Theorem 1.13 can be summarized as follows: Choose a small γ > 0. Consider a G with δ(g) > ( γ) V (G). Choose 0 < ε γ. By the Regularity Lemma there is an ε-regular partition of V (G). In the resulting 2-colored reduced graph R we have δ(r) 3 4 V (R) (by Proposition 3.2). Applying Lemma 4.1 to R, obtain a perfect matching in R for which the red edges are contained in a red component and the blue edges are contained in a blue component. The perfect matching in R corresponds to ε-regular pairs in G. Since the red part of the matching lies in a red component of R, the red pairs in G may be connected by red paths; very few vertices are discarded in the connection process. A similar thing may be done for the blue part of the matching in G. Very few vertices may be discarded to obtain super-regularity in the matching pairs in G; by the Blow-up Lemma, there is in each pair a monochromatic spanning path from the endpoints of the paths in the previous step. The result is a red cycle and a blue cycle that span all but very few vertices: the ones left out from the ε-regular partition, the ones which were necessarily discarded as a result of taking paths between pairs of partition sets, and the ones discarded to obtain super-regularity. So, by starting with a slightly higher minimum degree and applying the Regularity Lemma, minimum degree 3n 4 is preserved in the reduced graph. By an independent lemma we obtain a structure which, when analyzed in the original graph, yields the nearly-spanning monochromatic cycles. Assuming that proving Conjecture 1.12 is the goal, the result must be strengthened to have the cycles span the entire vertex set and the minimum degree condition must be weakened to exactly 3n 4. Achieving the former seems difficult, seeing as little (if anything) is known about the vertices not already spanned by the constructed cycles. Doing the latter will cause the proof to fail, since the regularity graph might then have minimum degree slightly less than 3n 4 (which means that Lemma 4.1 does not apply). Addressing the first difficulty is the substance of the proof of Theorem 1.14; dealing with the second difficulty is the topic of Section 7. 5 Our Proof The following summarizes our proof, the new concepts of which will be introduced in the subsequent subsections: Choose a small γ > 0. Consider a G with δ(g) ( γ) V (G). Analyze the structure of G by applying Lemma 5.1(I). In the largest robust component of each color, set aside an absorbing structure. Apply the Balogh, et al. argument to obtain the matching which gives the nearly spanning cycles, with two differences: 1) Apply Lemma 5.1(II) instead of Lemma 4.1, and 2) when connecting the ε-regular pairs in G by monochromatic paths, also connect the path from each absorbing structure so that the absorbing paths are embedded into the nearly spanning cycles. Use the absorbing property of the absorbing structures to absorb all of the unspanned vertices to obtain a Lehel partition of G. 9

10 5.1 Robust Components The idea of a robust component is that it is in no way easily separable; whereas a component might not be 2-edge-connected, a robust component should not have any sparse cuts. Thus a robust component should be highly connected and have a substantial minimum degree: Definition 5.1. Let 0 < α < η and let G be a graph on n vertices. A set X V (G) is (η, α)-robust if δ(g[x]) ηn and there is no partition {X 1, X 2 } of X which is an α-sparse pair in G. 13 Intuitively, one would suppose that a graph with minimum degree 3n 4 has a spanning robust component. However, a 2-colored graph with minimum degree 3n 4 will have non-obvious red and blue robust component structures even if the component structure is known. Hence we need an analog of Lemma 4.1: Lemma 5.1 (DeBiasio, N ). Let G be a graph with δ(g) (3/4 + γ)n. (I) In every 2-coloring E(G 1 ) E(G 2 ) = E(G) there exist robust subgraphs H 1 G 1, H 2 G 2 such that one of the following holds: (i) H 1 or H 2 spans V (G) (ii) H i (3/4 + γ/2)n for i = 1, 2 and V (G) = V (H 1 ) V (H 2 ). (II) H 1 H 2 contains a perfect matching. But why consider robustness? The main point of robustness is that within a robust component we have substantial connectedness amongst all vertices; if we have enough connectedness on a large enough graph, we should have a subset of the robust components that is very specifically connected to any disjoint subset of small size. (The specific connectedness is described in the next subsection.) We define a property that implies precisely this: Definition 5.2. Let G be a graph on n vertices. We say G has the (k, α)-connecting property if for all x, y V (G), con k (x, y) (αn) k, where con i (x, y) is the set of x, y-paths having i internal vertices. There is a direct relationship between robustness and having a connecting property; robust components have a connecting property, and a subset with a connecting property must be in some sense robust. Lemma 5.2 (DeBiasio and N ). Let 0 < α η 1 and let G be a graph on n vertices. (i) If G is (η, α)-robust, then G has the (j, α 2 )-connecting property for some j 1 α. (ii) If δ(g) ηn and G has the (k, α)-connecting property, then G is (η, α k+1 ) robust. 5.2 Absorbing Properties In what way would we want a subset (say A) of a robust component to be conveniently connected to any disjoint subset of small size? Consider what would be desirable if, after applying the argument of Balogh, et al., we wanted to manually insert the leftover vertices into the nearly spanning red and blue cycles. If a leftover vertex v is adjacent in blue to two consecutive vertices on the blue 13 For the definition of a sparse cut, see [6]. 10

11 cycle, then we may easily extend the blue cycle to include v. (See Figure 4.) If we could manage this one-by-one for all leftover vertices, then we would be happy. But without any information about the leftover vertices, this seems quite difficult unless we have pre-arranged for these leftovers to be in many red or blue triangles that are partially-embedded into the nearly spanning cycles. Figure 4: ab acts as an absorbing gadget for v. As it turns out, triangles are not the only substructures that can help this process of absorbing leftover vertices; any odd cycle will do, if it is correctly embedded into the monochromatic cycle. (See Figure 5.) So an adequate absorbing set in a robust component H is a subset A such that for every set B H \ A of size βn and each v B, for some k there are many 2k-sets in A forming a (2k + 1)-cycle with v. Fortunately, this follows from the definition of robustness except when the robust component is nearly bipartite. Figure 5: Since xv 1 v 2...v 10 is a cycle, the path v 2 v 1...v 4 v 3...v 6 v 5...v 8 v 7...v 9 v 10 acts as an absorbing gadget for x. Definition 5.3. Let G be a graph on n vertices. We say G has the α-vertex-absorbing property if for all v V (G) there exists i 1 α such that v is contained in at least (αn)2i cycles of length 2i + 1. Definition 5.4. Let G be a graph on n vertices. We say G is (η, α)-near-bipartite if there exists X V (G) such that (i) e(x) < αn 2, (ii) e(v (G) \ X) < αn 2, (iii) δ(x, V (G) \ X) ηn, (iv) δ(v (G) \ X, X) ηn. 11

12 Proposition 5.3 (DeBiasio and N ). Let 0 < α η 1. If G is (η, α)-robust, then either G has the α 2 -vertex-absorbing property or G is (α, η/2)-near bipartite. When a robust component is nearly bipartite, we are not guaranteed that there are many odd cycles. There are, however, even cycles which if correctly embedded in the nearly spanning monochromatic cycles could absorb pairs of vertices. (See Figure 6.) By finding a similar type of absorbing set with even cycles, we can absorb pairs of leftover vertices: one vertex from each part of the near-bipartition. x y Figure 6: A bipartite absorbing gadget that can absorb x and y. Definition 5.5. Let G be a graph on n vertices. Let H = G[X, Y ] be a bipartite subgraph of G. We say that H has the α-pair-absorbing property if for all x X, y Y there exists i 1 α such that x and y are contained in at least (αn) 2i cycles of length 2i + 2 (note that there are an even number of vertices between x and y on each path between x and y on such a cycle). Proposition 5.4 (DeBiasio and N ). Let 0 < α η 1. If G is (η, α)-robust and (α, η/2)- near bipartite, then G has the α 2 -pair absorbing-property. Hence within robust components, we must have either a vertex-absorbing property or we have a nearly-bipartite situation with a pair-absorbing property. So in a robust component, a nearly spanning cycle with properly embedded absorbing gadgets is as good as a spanning cycle. 5.3 Absorbing Structures To move from a nearly-spanning cycle to a spanning cycle, we must ensure that many absorbing gadgets are properly embedded into the nearly spanning cycle. In order to do this in our proof, before applying the Balogh, et al. argument we build an absorbing path in each color by properly linking the absorbing gadgets together. Then when connecting the ε-regular pairs with monochromatic paths, we also connect in the absorbing path. (See Figure 7.) Because the absorbing set lies in a robust component, which has a connecting property, we are able to link the substructures from our absorbing set into gadgets. By the same token, we are able to link the gadgets into one path by finding short paths between their endpoints. This gives us our monochromatic absorbing paths that will each be a segment of the nearly spanning monochromatic cycles. (See Figure 8.) In each color, the absorbing path together with a small set of vertices (called buffer vertices ) form the absorbing structure for that color. We then apply 12

13 Figure 7 the regularity argument to the graph with the absorbing structures removed. 14 Proceeding with the proof, we absorb all of the leftover vertices and the buffer vertices into the nearly spanning cycles. (a) The absorbing set of (2i)-sets that form (2i + 1)-cycles with leftovers. (b) The absorbing gadgets formed by connecting each (2i)-set in a certain way. (c) The absorbing path formed by linking together absorbing gadgets. Figure Buffer Vertices and the Near-Bipartite Case What is the purpose of buffer vertices? Consider this situation: H 1 (say the red robust component) is nearly-bipartite, and there are more leftover vertices in one part of H 1 than the other part. After absorbing as many pairs of leftover vertices in H 1 into the red cycle as possible, there are still some leftovers in one part of H 1. If these vertices are in H 2 and H 2 has the vertex-absorbing property, then no problem arises; but if not, then we are stuck. This is the purpose of the buffer vertices: to be conveniently-placed leftovers so that the absorbing process does not get stuck. 14 Since robust components are not sensitive to small deletions, this does not disrupt the robust component structure. 13

14 So, how do we choose the buffer vertices? If the largest robust component in a color is not nearly-bipartite, then it has the α 2 -vertex-absorbing property and we choose the buffer vertices of that color to be the empty set. (Having the α 2 -vertex-absorbing property in a color means that we can absorb all of the leftover vertices in the largest robust component of that color.) However, if the largest robust component of a color (say red) is nearly-bipartite, then we must make a non-trivial selection of the buffer vertices: we take some 15 vertices from the smaller part of the near-bipartition, and twice as many vertices from the intersection of the larger part of the near-bipartition and the largest robust component of the other color (blue). From analysis of the robust component structure in each color, we know that our selection is possible: Lemma 5.5. Let 0 < α η γ 1 and let G be a graph of order n 1 such that δ(g) ( 3 4 +γ)n and let E(G 1 ) E(G 2 ) = E(G) be a 2-edge coloring of G. Suppose there exists i [2] such that H i is (η, α)-robust and contains a (2η, 2α)-robust subgraph H i. If H i is α 2 -near-bipartite where {U 1 i, U 2 i } is a partition of U i := V (H i ) which witnesses H i being (η, α)-near-bipartite ( U 1 i U 2 i ), then H 3 i is not α-near-bipartite and U 1 i \ U 3 i αn. So, after applying the Balogh, et al. argument and obtaining the nearly-spanning cycles, there are more leftovers in the larger part of the red near-bipartition than in the smaller part. By the red α 2 -pair-absorbing property, absorb the pairs in red until there are only leftovers in both red and blue, and then (by the α 2 -vertex-absorbing property) absorb the leftover-leftovers in blue. Then we are done. The absorbing method is an interesting and recently developed tool. The details of the method in our proof are too complicated to provide here see [6]. A simpler but still illustrative application of the method is given in the next section. 6 An Example of the Absorbing Method We consider and prove the following: Statement 6.1. Let 0 < ε ρ α η 1 n and G be an (η, α)-robust graph on n vertices with the following properties: For all A, B, X V (G) such that A, B ηn and X ρn, G \ X contains a path with the first endpoint in A \ X, the second endpoint in B \ X, and spanning all but εn vertices of G \ X. For all u V (G), u is in at least (αn) 4 5-cycles. G is Hamiltonian. 6.1 Outline of Proof The condition that gives a nearly-spanning path implies a nearly-spanning cycle C; to guarantee a Hamiltonian cycle, we must insert the remaining εn vertices into C. As we have seen from 15 An amount determined by the absorbing properties: a fourth of the maximum size of a subset guaranteed to be absorbed. By careful parameter selection, we also guarantee that the number of leftover vertices from regularity and blow-up is at most this amount. 14

15 previous discussion about absorbing gadgets, the 5-cycle condition is helpful here; if we can embed an absorbing structure into C, then we can insert the leftover vertices. (a) A 5-cycle. (b) How the 5-cycle structure could help absorb. Figure 9 (c) The 4-set that becomes an absorbing gadget. Outline (See Figure 10): We want an absorbing path P that can absorb up to εn leftovers. The first step in getting P is finding a collection of disjoint 4-sets A ( ) V (G) 4 with the following property: any B V (G) with B < εn can be absorbed by A. 16 To find A we will use Lemma 6.2. The next step in getting P is to connect the 4-sets of A into a path of specific order. To do this, we will use the connecting property that follows from the robustness of G. This path, on at most ρn vertices, is P. The endpoints of P have neighborhoods large enough that there is a path ˆP from one neighborhood to the other that spans all but εn leftovers. ˆP P is a nearly-spanning cycle with properly embedded gadgets. We absorb the leftover vertices and obtain a cycle that spans G. 6.2 Proof We begin with the following lemma: Lemma 6.2. Let 0 < ρ < 1. Let S = ( ) [n] 4 and let T = [n]. There exists n0 such that the following holds: If n n 0 and Γ is an S, T -bipartite graph having the property δ(t, S) 2ρn 4, then there exists a collection of disjoint sets A S such that A A A ρn, δ(t, A ) ρ 2 n, and δ(a, T ) 1. Furthermore, for all B T with B ρ 2 n, Γ[A, B] contains a matching saturating B. Proof. (of Lemma 6.2) We will show that a randomly chosen subset of S will almost surely satisfy all the properties that A must satisfy. Then by deleting some elements from the randomly chosen set, we will obtain the actual set A which has all of the desired properties. Set p = 3ρ. Let A be a randomly chosen subset of ( ) [n] n 3 4 where each each element is chosen independently with probability p. We note several basic properties of A (due to the Chernoff inequality unless otherwise indicated): 16 In precise terms, there is an injection ψ : B A such that {b} ψ(b) forms a 5-cycle for each b B. 15

16 (i) With probability 1 exp{ n/ log n} we have ( ) n A 2p 4 2 3ρ n 3 n4 24 n = ρn 4 and thus A A A 4 A ρn. (ii) Let A A = {(S 1, S 2 ) A A : S 1 S 2 }. Then ( ) n E [ A A ] 4 p 4 ( n 3 ) p 4 ρn 8 ρ 2 = ρ2 n 4 So by Markov s inequality, [ ] Pr A A ρ2 n ; (iii) δ(t, S) 2ρn 4. So with probability 1 exp{ n/ log n} we have for u T, deg(u, A) 1 2 p(2ρn4 ) = 3ρ 2 n. Let A be a subset of S for which properties (i) (iii) hold. Now, in every pair of intersecting sets (S 1, S 2 ) in A, delete one of S 1 or S 2 ; let A be the resulting set. By properties (ii) and (iii), we have deg(u, A ) deg(u, A ) 4 A A 3ρ 2 n 2ρ 2 n = ρ 2 n Let A A be a maximal subset having the property that δ(a, T ) 1 and note that by maximality we still have deg(u, A ) 2ρ 2 n. So for all B T with B 2ρ 2 n, we can greedily choose a matching in Γ[A, B] which saturates B. Now, to proceed with the proof of the statement: Proof. (of Statement 6.1) Let ρ = α 4 and ε = α 10. We apply Lemma 6.2 with S = ( ) V (G) 4, T = V (G), using (ρα) as the ρ in the lemma, and letting {S, t} E(Γ) iff S {t} forms a 5-cycle in G. Since each vertex of G is on at least α 4 n 4 5-cycles, δ(t, S) α 4 n 4 = ρn 4 (ρα)n 4 and by the lemma there is a collection of disjoint 4-sets of A V (G) of size at most (ρα)n 4 such that any B V (G) with B < (ρα) 2 n can be absorbed by A. Now we will show how to turn the set A = P into the desired path P. P A Give A an ordering {P i } l i=1. Now, each element of A is a path P i = a i b i c i d i on 4 vertices; give A the ordering b 1 a 1 c 1 d 1 b 2 a 2 c 2 d 2...b i a i c i d i...b l a l c l d l. By Lemma 5.2, G has the (j, α 2 )-connecting property for some j 1 α. We use this property to build disjoint paths with internal vertices in G \ A from a i to c i for i [l] and from d i to b i for i [l 1] (see Figure 10(b)). Call the resulting path P. In building P, we used 2l 1 paths which each used j internal vertices from outside A. So P = (2l 1)j + A < 2 ραn 4 1 α + ραn = ρn 2 + ραn < ρn. 16

17 Now, since G is (η, α)-robust, d(b 1 ), d(d l ) ηn. So N(b 1 ), N(d l ) ηn. Hence there is a path ˆP from N(b 1 ) \ P to N(d l ) \ P in G \ P that spans all but at most εn = α 10 n = (ρα) 2 n vertices. So ˆP P is a cycle that spans all but at most (ρα) 2 n vertices of G. Now by Lemma 6.2, there is an injective mapping ϕ : V (G) \ ( ˆP P ) [l] such that va ϕ(v) b ϕ(v) c ϕ(v) d ϕ(v) is a 5-cycle. For each v V (G) \ ( ˆP P ) we may extend the segment b ϕ(v) a ϕ(v) c ϕ(v) d ϕ(v) to b ϕ(v) c ϕ(v) a ϕ(v) vd ϕ(v). This extension of ˆP P is a cycle spanning G. (a) The absorbing set A, a collection of 4-sets. (b) P (c) ˆP P, plus leftovers. (d) Leftovers are highly connected to P. Figure 10 (e) Leftovers absorbed. 7 An Exact Result To prove Conjecture 1.12 for large n, we must weaken the minimum degree condition to exactly 3n 4. There are some points at which the argument given in Section 5 no longer holds with this weakened condition: most notably, the reduced graph will not necessarily have minimum degree 3n 4. This means that the reduced graph may not have the structure or perfect matching guaranteed by Lemma 5.1. In this case, however, we know that the structure of the reduced graph (and therefore 17

18 the structure of the original graph) must look quite specific. In this sense we are operating under the stability method; we proceed as normal unless we have certain extremal conditions which prevent the standard procedure, in which case we directly analyze the graph. Extremal Conditions (1) and (2) are given after the sketch of the proof. 7.1 Sketch of Proof for Exact Result Consider a graph G of large order with δ(g) 3 4 V (G). Analyze the structure of G: We either have the conclusion of Lemma 5.1(I), or G satisfies Extremal Conditions (1). If the latter, by Lemma 7.1 G admits a Lehel partition. If the former, proceed as usual: In the largest robust component of each color, set aside an absorbing structure. Choose 0 < ε γ 1. By the Regularity Lemma there is an ε-regular partition of V (G). In the resulting 2-colored reduced graph R we have δ(r) ( 3 4 γ) V (R) (by Proposition 3.2). If in R we have a perfect matching in H 1 H 2 (where H 2 is some robust component in G 2), then we proceed. Otherwise, G satisfies Extremal Conditions (2) and by Lemma 7.1 G admits a Lehel partition. The perfect matching in R corresponds to ε-regular pairs in G. We may apply our standard connecting procedure to the pairs and absorbing paths, and discard few vertices to obtain spanning paths within the pairs. So we have nearly spanning monochromatic cycles with conveniently embedded absorbing gadgets. Use the absorbing structures to absorb all of the unspanned vertices to obtain a Lehel partition of G. 7.2 Extremal Cases Definition 7.1. Given 0 < α 1, we say that G satisfies Extremal Conditions (1) if there exist sets X, Y V (G) with (1/2 α)n X, Y (1/2 + α)n such that e 1 (X, V (G) \ X) < αn 2 and e 2 (Y, V \ Y ) < αn 2. Definition 7.2. Given 0 < α 1, we say that G satisfies Extremal Conditions (2) if there exist disjoint sets X 1, X 2, Y V (G) with (1/4 α)n X 1, X 2 (1/4 + α)n and (1/2 α)n Y (1/2 + α)n such that e 1 (X 1, Y X 2 ) < αn 2, e 2 (X 1 ) < αn 2, e 2 (X 2, Y X 1 ) < αn 2, and e 1 (X 2 ) < αn 2. It follows from the definitions of the extremal conditions that if G satisfies Extremal Conditions (1), then G looks like Figure 2, and that if G satisfies Extremal Conditions (2), then G looks like Figure 11. In these cases, we have a very specific structure of G which allows us to find a Lehel partition directly: Lemma 7.1 (DeBiasio and N ). Given 0 < α 1 and a 2-colored graph G on n vertices with δ(g) 3n 4, if G satisfies Extremal Conditions (1) or (2) then G admits a Lehel partition. Proving Lemma 7.1 is a matter of leveraging the minimum degree condition to find guaranteed edges between certain parts. In Extremal Conditions (1), G must have a blue matching of size 2 across the blue sparse cut or a red matching of size 2 across the red sparse cut. In Extremal Conditions (2), G must have a monochromatic matching of size 2 across X 1 and X 2. In both 18

19 Figure 11 cases, we use the monochromatic matchings of size 2, together with the nearly complete parts or edge-cuts, to manually construct a Lehel partition of G. References [1] P. Allen. Covering two-edge-coloured complete graphs with two disjoint monochromatic cycles. Combin. Probab. Comput., 17 (2008), no. 4, [2] J. Ayel. Sur l existence de deux cycles supplémentaires unicolores, disjoints et de couleurs différentes dans un graph complet bicolore. PhD thesis, Université de Grenoble (1979). [3] J. Balogh, J. Barát, D. Gerbner, A. Gyárfás and G. Sárközy. Partitioning 2-edge-colored graphs by monochromatic paths and cycles. To appear in Combinatorica. [4] F. Benevides, T. Luczak, A. Scott, J. Skokan and M. White. Monochromatic cycles in 2- coloured graphs. Combin. Probab. Comput., 21 (2012), [5] S. Bessy and S. Thomassé. Partitioning a graph into a cycle and an anticycle, a proof of Lehel s conjecture. J. Combin. Theory Series B 100 (2010), no. 2, [6] L. DeBiasio and L. Nelsen. Covering 2-edge-colored graphs with a pair of cycles. Manuscript. [7] P. Erdős, A. Gyárfás and L. Pyber. Vertex coverings by monochromatic cycles and trees. J. Combin. Theory Ser. B, 51 (1991), no. 1, [8] R. Faudree, L. Lesniak and I. Schiermeyer. On the circumference of a graph and its complement. Disc. Math. 309 (2009) [9] L. Gerencsér and A. Gyárfás. On Ramsey-type problems. Ann. Sci. Budapest. Eötvös Sect. Math, 10 (1967), [10] A. Gyárfás, M. Ruszinkó, G. Sárközy and E. Szemerédi. An improved bound for the monochromatic cycle partition number. J. Combin. Theory Ser. B, 96 (2006),

20 [11] A. Gyárfás, M. Ruszinkó, G. Sárközy and E. Szemerédi. Partitioning 3-colored complete graphs into three monochromatic cycles. Elec. J. Combin., 18, no. 53, (2011). [12] A. Gyárfás and G. Sárközy. Star versus two stripes Ramsey numbers and a conjecture of Schelp. Combin. Probab. Comput., 21 (2012), [13] A. Gyárfás. Vertex coverings by monochromatic paths and cycles. J. Graph Theory, 7 (1983), [14] A. Gyárfás. Covering complete graphs by monochromatic paths. Irreg. of Part., Alg. and Combin., 8 (1989), [15] P. Haxell and Y. Kohayakawa. Partitioning by monochromatic trees. J. Combin. Theory Ser. B, 68 (1996), [16] D. Kühn and D. Osthus. Embedding large subgraphs into dense graphs. In: Surveys in Combinatorics, in: London Math. Soc. Lecture Note Ser., 365 (2009), [17] J. Komlós and M. Simonovits. Szemerédis regularity lemma and its applications in graph theory. In: Combinatorics, Paul Erdős is Eighty, 22 (1993), in: Bolyai Soc. Math. Stud., 2, (1996), [18] H. Li, V. Nikiforov and R. Schelp. A new type of Ramsey-Turán problems. Disc. Math., 310 (2010), [19] T. Luczak, V. Rödl and E. Szemeredi. Partitioning two-colored complete graphs into two monochromatic cycles. Combin. Probab. Comput., 7 (1998), [20] V. Nikiforov and R. Schelp. Cycles and stability. J. Combin. Theory Ser. B, 98 (2008), [21] A. Pokrovskiy. Partitioning edge-coloured complete graphs into monochromatic cycles and paths. arxiv: v1. [22] V. Rödl, A. Ruciński and E. Szemerédi. Perfect matchings in large uniform hypergaphs with large minimum collective degree. J. Combin. Theory Ser. A 116 (2009), no. 3, [23] R. Schelp. Some Ramsey-Turán type problems and related questions. Disc. Math., 312 (2012), no. 14,