Jacques Verstraëte

Transcription

1 2 - Turán s Theorem Jacques Verstraëte jacques@ucsd.edu 1 Introduction The aim of this section is to state and prove Turán s Theorem [17] and to discuss some of its generalizations, including the Erdős-Stone Theorem [7], Simonovits Theorem on color-critical graphs [14] and the stability method, and the Andrásfai- Erdős-Sós Theorem. In the following material, we use the notion of an independent set of a graph G: this is a set of I vertices of G such that no two vertices of I are adjacent in G. These are sometimes referred to as stable sets in the literature. The chromatic number of a graph G, denoted χ(g), is the smallest r such that V (G) has a partition into independent sets V 1, V 2,..., V r. In other words, we may assign r colors to the vertices of G in such a way that no two vertices of the same color r adjacent. Such graphs are also called r-partite. The graph G is a complete r-partite graph if all edges between V i and V j are present for i, j : 1 i < j r. We say G is a balanced r-partite graph if V 1 V 2 V r V If X, Y are sets of vertices in a graph, let e(x, Y ) denote the number of edges {x, y} with x X and y Y, and e(x) = e(x, X). If f, g : N R + R + are functions, we write f = O δ (g) if there exists a constant c(δ) depending only on δ such that f(n, δ) c(δ)g(n, δ) for all n N and for all δ R +. Similarly f = Ω δ (g) if and only if g = O δ (f), and f = Θ δ (g) if f = O δ (g) and g = O δ (f). If f = Θ(g), we say that f and g have the same order of magnitude. We write ex(n, F) instead of ex 2 (n, F) when F is a family of graphs. 1

2 1.1 Mantel s Theorem The first theorem of extremal graph theory is Mantel s Theorem [12], which shows that the extremal triangle-free graphs are balanced complete bipartite graphs. There are many proofs of this result; we give the inductive proof (by deleting vertices). Theorem 1. [12] If G is a triangle-free n-vertex graph, then e(g) n 2 /4, with equality if and only if G is a balanced complete bipartite graph. Proof. We proceed by induction on n. The theorem is trivial if n = 1, 2. For n 3, let G be a triangle-free n-vertex graph, and suppose e(g) n 2 /4. We claim G is a balanced complete bipartite graph. Since adding any edge to a balanced complete bipartite graph gives a triangle, it is sufficient to prove the theorem for any graph G with e(g ) = n 2 /4 obtained by deleting edges of G. By Proposition 1, G contains a vertex v with d G (v) n/2, since (n/2)( n/2 + 1) > n 2 /4 for all n 1. Let G = G {v}. Then n e(g 2 n ) = 4 2 (n 1) 2 By induction, G is a balanced complete bipartite graph with parts say X 0 and X 1, and e(g ) = (n 1) 2 /4. This also implies d G (v) = n/2. Now for some i {0, 1}, N G (v) X i =, otherwise G contains a triangle containing v. Therefore N G (v) X 2 i. However then X 2 i N G (v) n/2, and so X i n/2. We conclude G is bipartite, with parts X i {v} and X 2 i. Furthermore, e(g ) = n 2 /4 and e(g ) X 1 X 2 implies X 1 n/2 and X 2 n/2, so G is balanced. 4. In the language of Turán Numbers, this theorem says ex(n, K 3 ) = n 2 /4 and π(k 3 ) = Turán s Theorem The proof of Mantel s Theorem generalizes to give Turán Numbers for complete graphs. This was proved by Turán [17]. We define the Turán Graphs to be n- vertex graphs T r (n) which admit a vertex partition V 1 V 2 V r such that V 1 V 2 V r V such that T r (n) contains all edges between V i and V j, for 1 i < j r and no edges in any V i. We call this the r-partition of T r (n), 2

3 and T r (n) is the balanced complete r-partite graph. Then e(t r (n)) = n + i n + j. r r 0 i<j<r In the special case r = 2, we get the balanced complete bipartite graph. Some simple observations on T r (n) are that T r (n 1) is obtained from T r (n) by deleting a vertex of T r (n) of lowest degree, namely a vertex of degree δ(t r (n)) = n/r. In fact, T r (n) is the n-vertex graph with highest minimum degree amongst all n-vertex graphs with e(t r (n)) edges. Also, amongst all r-partite n-vertex graphs, T r (n) has the maximum number of edges, and if G is an r-partite n-vertex graph with e(g) = e(t r (n)), then G = T r (n). Note also that e(t r (n)) (1 1 r )( n 2). Here is Turán s Theorem: Theorem 2. [17] Let G be a K r+1 -free n-vertex graph. Then e(g) e(t r (n)) with equality if and only if G = T r (n). Proof. Proceed by induction on n. The cases n r follow from the fact that e(t r (n)) = e(k n ) for n r. For n r + 1, let G be an n-vertex K r+1 -free graph with e(g) e(t r (n)). We may assume e(g) = e(t r (n)) by deleting edges and noting T r (n) is a maximal K r+1 -free graph. Since T r (n) is the graph with e(t r (n)) edges of highest minimum degree, δ(g) δ(t r (n)). Removing a vertex v of degree δ(g) from G, we get a graph G with e(g ) e(t r (n)) δ(t r (n)) = e(t r (n 1)). It follows that G = T r (n 1). If V 1 V 2 V r is the r-partition of T r (n 1), and v is adjacent to at least one vertex v i V i for all i r, then {v, v 1, v 2,..., v r } is the vertex set of K r+1 in G, a contradiction. Therefore some V i contains no neighbors of v. Then the V j : j i together with V i {v} form an r-partition of G, so G = T r (n). This theorem shows π(k r ) = 1 1 r 1 for r 2. We will show how to adapt the proof above to determining π(f) for every family F of graphs. 2 The Erdős-Stone Theorem To determine π(f) for every family F of graphs, we use the key fact that T r (n) contains every graph F with χ(f ) r and V (F ) n, and contains no graph F with χ(f ) > r. Let χ(f) = min{χ(f ) : F F}. The following result is referred to as the Erdős-Stone Theorem [7]: 3

4 Theorem 3. [7] For every r, m 1, π(t r (m)) = 1 1 r 1 = π(k r). In particular, for any family F of graphs with χ(f) 2, π(f) = 1 1 χ(f) 1. Proof. Let ε > 0 and let G be an n-vertex graph with density π(k r ) + ε. By the supersaturation lemma and Turán s Theorem, G contains at least δn r copies of K r for some δ > 0 depending only on ε. Let H be the hypergraph with V (H) = V (G) and E(H) = {V (K r ) : K r G}. Then H is an r-graph with at least δn r edges. By Theorem 14, with t = m/r, H contains K t:r provided n is large enough, and this implies T r (m) G. Therefore π(k r ) π(t r (m)) π(k r ) + ε for any ε > 0, and so π(t r (m)) = π(k r ). The second part of the theorem was observed by Simonovits: if F F has χ(f ) = r = χ(f), then F T r (m) where m r V (F ). This shows π(f) π(t r (m)). Since π(f) π(k r ), we are done. This theorem shows that the Turán densities for graphs lie in the set {0, 1 2, 2 3, 3 4,... }. 2.1 Improvements on Erdős-Stone It is useful to ask for quantitative bounds in the Erdős-Stone Theorem. Specifically, for r 2 and > 0, let f r (n, ) = max{m N : ex(n, T r (rm)) ex(n, K r ) n2 1}. The original proof of the Erdős-Stone Theorem [7] gives where the iterated logarithm is r times. f r (n, ) = Ω,r ( log log log... log n), This turns out to be very far from the truth, and gives very poor bounds on the speed of the convergence of ex(n, F) to π(k r ) when χ(f) = r. We now briefly outline historical improvement, and later we will show how to determine f r (n, ) quite accurately. For r = 2, the Kövari-Sós-Turán Theorem (see Lecture Notes 1) shows for all n, ex(n, T 2 (2m)) = ex(n, K m,m ) 1 2 (m 1)1/m n 2 1/m + 1 (m 1)n. 2 4

5 From this the reader can gather via asymptotic calculations that for any > 0, f 2 (n, ) log 1 n. We shall shortly see that this is asymptotically best possible up to a factor two, as proved by Bollobás and Erdős [3, 4]. It turns out to be very difficult to give explicit constructions of graphs which show f r (n, ) is small. For the first time in our work, we appeal to random graphs for the following strong theorem of Bollobás and Erdős [3, 4], which determines f 2 (n, ) up to an asymptotic factor of two. Theorem 4. [3] For each r 1 and 0 < < 1/r 2, f r+1 (n, ) 2 log 1 r 2 n. In particular, if = o(1) then f r (n, ) 2 log 1 n for all r 2. Proof. Let T be the Turán graph T r (n). If we add a K t,t -free graph H in some part of T r (n), then we obtain a graph G that does not contain T r+1 ((r + 1)t) and has e(t r (n)) + e(h) = ex(n, K r ) + e(h) edges. If we can ensure e(h) 1 2 n2 1, then f r+1 (n, ) t. Taking V (H) = n/r, we would like H to be a K t,t -free graph with e(h) 1 2 n2 1, in other words, t f 2 ( n/r, r 2 ). This certainly shows for all r 2 that f r+1 (n, ) f 2 ( n/r, r 2 ) provided < 1/r 2. Now we show f 2 (n, ) 2 log 1 n. To do this, we construct a random graph G on n vertices, where each edge is present independently with probability. Then E(e(G)) = ( n 2). Let Y be the number of Kt,t G where t = 2 log 1 n. Then ( ) n E(Y ) = t2 2 t 1 < ( t n 2 ) t. 2t The definition of t ensures t n 2 < 1 and therefore E(Y ) < 1. Let G be obtained from G by deleting one edge from every copy of K t,t G. Then we obtain a graph G such that E(G ) E(e(G) Y ) > ( ) n 1. 2 Then G is a K t,t -free graph with more than ( n 2) 1 edges, so f2 (n, ) 2 log 1 n. We conclude for < 1/r 2 that This completes the proof. f r+1 (n, ) f 2 ( n/r, r 2 ) 2 log 1 r 2 n. 5

6 Bollobás and Erdős [3, 4] determined the order of magnitude of f r (n, ) for all r 2, namely f r (n, ) = Θ r, (log n). Chvátal and Szemerédi [6] determined the dependence on of the implicit constant to be 1/ log(1/), and finally, Ishigami [9] determined the asymptotic behavior of f r (n, ) for all r 3 conditional on the asymptotic behavior of f 2 (n, ): Theorem 5. Let r 2 and let (n) = = o(1) as n. Then f r (n, ) f 2 (n, ) and, in particular, log 1 n f r (n, ) 2 log 1 n. The proof of Theorem 5 in [9] uses Szemerédi s Regularity Lemma and is rather technical. 3 The stability method We next turn to situations where extremal numbers can be determined exactly. The idea is to use the stability method, which in rough terms says that we first determine the approximate structure of near extremal graphs, and then use this approximate structure to obtain exact structure of exactly extremal graphs. A key lemma in the stability approach is as follows: Lemma 6. Let F be a graph. Suppose that for all n n 0, if G n is an extremal n-vertex F -free graph, then e(g n 1 ) = e(g n ) δ(g n ), and if H n is an n-vertex F -free graph with δ(h n ) δ(g n ) and e(h n ) e(g n ), then H n = G n. Then for n n 2 0, if J is any n-vertex graph with e(j n) e(g n ), then J = G n. Proof. If δ(j n ) δ(g n ), then we are done. Suppose δ(j n ) < δ(g n ). Remove from J n a vertex of smallest degree to obtain a graph J n 1. If δ(j n 1 ) < δ(g n 1 ), repeat. Continue until δ(j m ) δ(g m ) or m = n 0 1. If m n 0, then δ(j m ) δ(g m ) and e(j m ) e(g m ) implies J m = G m by assumption. Otherwise, m = n 0 and ( ) m e(j m ) e(g m ) + n m > m 2 m >. 2 This is a contradiction. 6

7 3.1 Color-critical graphs A good example of the application of the stability method is to color-critical graphs. We refer to a graph F as r-color critical if χ(f {e}) < χ(f ) = r for some e E(F ). For instance, complete graphs and odd cycles are color critical. Our aim is to show that if F is (r + 1)-color-critical, then ex(n, F ) = e(t r (n)) with equality only for T r (n). The following embedding lemma is required. An n-vertex r-partite graph G with parts Z 1, Z 2,..., Z r is almost complete if for every i r and j r with j i, every vertex z Z i has N G (z) Z j Z j. Lemma 7. Let F be a color-critical graph with χ(f ) = r+1, and let G be an almost complete r-partite graph with parts Z 1, Z 2,..., Z r. Then (i) (ii) If G is F -free, then the sets Z i are independent sets. If we add a vertex w to G and get an F -free graph, then w has at most V (F ) neighbors in some Z i. The proof of this lemma is straightforward. We now turn to the proof of the Simonovits Theorem [14] for color-critical graphs: Theorem 8. [14] Let r 2, and let F be an (r + 1)-color-critical graph. Then the unique extremal n-vertex F -free graphs are T r (n), provided n is large enough. Proof. Let G be an n-vertex F -free graph with at least e(t r (n)) edges and b = V (F ). Let V (G) = V. By Lemma 6, we may assume δ(g) δ(t r (n)) n n/r. By the Erdős-Stone Theorem, T r (rt) G where t = ω(1). Let V 1, V 2,..., V r be the parts of this T r (rt). By Lemma 7(i), each V i is an independent set. For each vertex w V, there exists a least i r such that w has less than b neighbors in V i, by Lemma 7(ii). We add w to V i in this case. Repeating this for every vertex w V, we obtain a partition W 1 W 2 W r of V with W i V i for all i r, and such that every vertex of W i has less than b neighbors in V i. Then since δ(g) n n/r, e(v i, V \V i ) (n n/r )t. On the other hand, e(v i, V \V i ) < b W i + e(w j, V i ) b W i + t W j. j i j i 7

8 Combining the last two inequalities, we have for every i r the inequality Since b = o(t), this implies W j + b t W i n n/r. j i W j n n/r. Since r i=1 W i = n, these inequalities imply W i n/r. Then e(w j, V i ) (n n/r)t. j i j i Since e(w j, V i ) tn/r for all i, j r, we conclude e(w j, V i ) tn/r W j V i for all i j and all j r. Let X i W i consists of all vertices w W i such that N j (w) V j t for all j i, noting V i X i. Since e(w i, V j ) W i V j for every j i, X i W i. Now the r- partite subgraph of G with parts X i and V j : j i is almost complete, so by Lemma 7(i), X i is an independent set. This holds for each i r, so every X i is independent. Let X = r i=1 X i and C = V \X. Then C = o(n) since X i W i n/r for i r. This implies that W = V \ X i has size o(n). Now if w W i \X i, then N G (w) X i < b. Therefore N G (w) j i X i > n n/r b W n n/r. We add w to X i in this case. Repeating this for every w W, we obtain a partition Y 1 Y 2 Y r of V, such that the Y i are the parts of an almost complete r-partite graph. By Lemma 7(i), each Y i is an independent set, and so G is r-partite. Since e(g) e(t r (n)), it follows G = T r (n). 3.2 Decomposition families If F is not color-critical, then the situation changes. The decomposition family of a graph F of chromatic number r,denoted M(F ), is the set of bipartite graphs obtained by deleting any set of r 2 color classes of F in any proper r-coloring of F. Then we can add ex(n, M(F )) edges into any part of T r 1 (n) and still not get a copy of F. For instance, if we take the graph F consisting of two triangles sharing one 8

9 vertex, then M(F ) consists of a pair of disjoint edges and a pair of edges sharing one vertex. Since ex(n, F) = 1 for all n 2, we deduce that ex(n, F ) n 2 /4 +1 and we can add one edge in any part of T 2 (n) to get an F -free construction. It is an exercise to show that this is tight for this particular graph F. In general, however, the family F will contain bipartite graphs and this leads us to return to the Degenerate Turán Problems. Simonovits [15] has shown that in fact the extremal F -free graphs in general when r + 1 = χ(f ) 3 are T r (n) plus some extremal F-free graphs pasted into the parts of T r (n). As another instance, if F is the Petersen graph, then the extremal graph (see Figure 1) was determined by Simonovits: take K m,n m where m = n/2 1 and add the complement of K n m 2 in the part of size n m. A survey of decomposition families is given in Simonovits survey paper [15]. Figure 1: The Petersen Graph 4 Andrásfai-Erdős-Sós Theorem In the last sections, we saw that an n-vertex graph with e(t r (n)) edges and no K r is r-colorable. It is not true, however, that all K r -free graphs are r-colorable. Let us examine the case r = 3. The first constructions of triangle-free graphs with arbitrarily high chromatic number were given by Zykov [18] and Mycielski [13]. A well-known construction comes from Kneser graphs. Consider the Kneser graphs K n t. Recall, for a set V with V = n, V (K n t ) = ( ) V t and E(K n t ) = {{S, T } : S, T ( V t ), S T = }. The Petersen graph shown in Figure 1 is isomorphic to K

10 A breakthrough theorem of Lovász [10] shows χ(k n t ) = n 2t + 2. If we take n = 3t 1, then K n t is a triangle-free graph with chromatic number t + 1. In other words, we have triangle-free graphs of arbitrarily large chromatic number. Hajnal showed how to add a large bipartite graph to K n t without creating triangles, whilst increasing the minimum degree to 1/3 o(1) of the number of vertices. The construction of triangle-free graphs of large chromatic number has received considerable attention, and we shall revisit this problem in the context of Ramsey Theory. Mantel s Theorem says that an n-vertex triangle-free graph with minimum degree at least n/2 is bipartite. The following theorem due, to Andrásfai, Erdős and Sós [2], gives the threshold minimum degree for a triangle-free graph to be bipartite. Theorem 9. [2] Let n 1. Then any n-vertex triangle-free graph of minimum degree larger than 2n/5 is bipartite. Furthermore, there exists for every n 5 a triangle-free non-bipartite graph of minimum degree 2n/5. Proof. Let G be such a graph. Suppose, for a contradiction, that G is not bipartite. Then there is a shortest odd cycle C G. This cycle C is an induced cycle in G, and if u, v V (C) have distance more than two on C, then N G (u) N G (v) =, since C is shortest. Since G is triangle-free, the same is true if u and v are adjacent on C. If C has length at least seven, then pick a vertex w and adjacent vertices u, v at distance at least three from w on C. Then N G (u), N G (v), N G (w) are disjoint, so N G (u) + N G (v) + N G (w) n. However N G (u) > 2n/5 and the same for v and w, so this is a contradiction. Therefore C has length exactly five. Suppose this cycle has vertices v 0, v 1, v 2, v 3, v 4 in that order on the cycle. Then with subscripts mod five,n G (v i ) N G (v i+1 ) =, which implies N G (v i 1 ) contains more than n/5 vertices in N G (v i+1 ) for all i {0, 1, 2, 3, 4}. If W i = N G (v i ) N G (v i+2 ) for i {0, 1, 2, 3, 4}, then the W i are pairwise disjoint sets of size more than n/5, a contradiction. For the construction for n 5, take a blowup of a cycle of length five (see Figure 2). In other words, let G be the graph with five parts V 0, V 1, V 2, V 3, V 4 where V 0 V 1 V 2 V 3 V 4 V 0 +1 and where V i and V i+1 are the parts of a complete bipartite subgraph of G for all i with subscripts mod five. Then G is triangle-free, and not bipartite. 10

11 Figure 2: The Blowup of a Pentagon 4.1 Chromatic threshold Brandt and Thomassé [5] showed that any n-vertex triangle-free graph of minimum degree more than n/3 is 4-colorable, whereas the afore-mentioned construction of Hajnal shows that there are n-vertex graphs G with δ(g) n/3 such that χ(g) = ω(1). The chromatic threshold of a graph H is the infimum of those δ [0, 1] such that every n-vertex H-free graph of minimum degree at least δn has bounded chromatic number. Thomassen [16] was the first to show that the chromatic threshold of K 3 is 1 3 confirmed in a strong sense in [5]. The chromatic threshold for K r for r 3 was determined to be (2r 5)/(2r 3) by Goddard and Lyle [8]. A novel approach to the chromatic threshold problem was given by Luczak and Thomassé [11] in terms of the VC-dimension of hypergraphs. The problem was completely solved by Allen, Böttcher, Kohayakawa and Morris [1] in the form of the following theorem. A graph H with χ(h) = r is r-near-acyclic if we can delete r 3 color classes from H to obtain a graph H such that V (H ) = X Y and X is an independent set and all edges of H in Y are disjoint. Theorem 10. [1] Let H be any graph with χ(h) = r 3. Then the chromatic threshold of H is an element of The chromatic threshold is not r 2 r 1 { r 3 r 2, 2r 5 2r 3, r 2 r 1 }. if and only if H is r-near- contains a forest, and the chromatic threshold is r 3 r 2 acyclic. if and only if the decomposition family M(H) 11

12 For instance, this theorem shows that odd cycles, the Petersen graph and the docecahedron graph (see Figure 3) have chromatic threshold zero, whereas the chromatic threshold of the icosahedron graph is 3/5. The proof of Theorem 10 is beyond the scope of this course, and makes very technical use of Szemerédi s Regularity Lemma. Figure 3: Dodecahedron Graph References [1] P. Allen, J. Bóttcher, S. Griffiths, and Y. Kohayakawa, The chromatic threshold of graphs Advances in Mathematics, 235 (2013) [2] B. Andrásfai, P. Erdős, and V. T. Sós, On the connection between chromatic number, maximal clique and minimal degree of a graph, Discrete Math. 8 (1974), [3] B. Bollobás and P. Erdős, On the structure of edge graphs, Bull. London Math. Soc. 5 (1973), [4] B. Bollobás, P. Erdős, and M. Simonovits, On the structure of edge graphs, II, J. London Math. Soc. (2) 12 (1976), [5] S. Brandt and S. Thomassé, Dense triangle-free graphs are four colorable: A solution to the Erdős-Simonovits problem. With S. Brandt, to appear. [6] V. Chvátal and E. Szemerédi, Notes on the ErdősStone Theorem, Ann. Discrete Math. 17 (1983),

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