RECAP: Extremal problems Examples

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1 RECAP: Extremal problems Examples Proposition 1. If G is an n-vertex graph with at most n edges then G is disconnected. A Question you always have to ask: Can we improve on this proposition? Answer. NO! The same statement is FALSE with n 1 in the place of n. Proposition 1 is best possible, as shown by P n. Proposition 1. + P n : The minimum value of eg over connected graphs is n 1. Proposition. If G is an n-vertex graph with at least n edges then G contains a cycle. Remark. Proposition is also best possible, e.g. P n. Proposition. + Remark: The maximum value of eg over acyclic i.e. cycle-free graphs is n 1. 1

2 RECAP: Extremal problems More example Vague description: An extremal problem asks for the maximum or minimum value of a parameter over a class of objects graphs, in most cases. Proposition. G is an n-vertex graph with δg n/, then G is connected. Remark. The above proposition is best possible, as shown by K n/ + K n/. Graph G + H is the disjoint union or sum of graphs G and H. For an integer m, mg is the graph consisting of m disjoint copies of G. Prop. + Remark: The maximum value of δg over disconnected graphs is n 1.

3 Extremal Problems graph graph type of value of property parameter extremum extremum connected eg min n 1 acyclic eg max n 1 n 1 disconnected δg max K 3 -free eg max non-hamiltonian δg max n 4 n 1 r-colorable eg max??? K r -free eg max??? 3

4 RECAP: Triangle-free subgraphs Theorem. Mantel, 1907 The maximum number of edges in an n-vertex triangle-free graph is n 4. Proof. i There is a triangle-free graph with n 4 edges. ii If G is a triangle-free graph, then eg n 4. Proof of ii is with extremality. Look at the neighborhood of a vertex of maximum degree. 4

5 Complete k-partite graphs A graph G is r-partite or r-colorable if there is a partition V 1 V r = V G of the vertex set, such that for every edge its endpoints are in different parts of the partition. G is a complete r-partite graph if there is a partition V 1 V r = V G of the vertex set, such that uv EG iff u and v are in different parts of the partition. If V i = n i, then G is denoted by K n1,...,n r. The Turán graph T n,r is the complete r-partite graph on n vertices whose partite sets differ in size by at most 1. All partite sets have size n/r or n/r. Lemma Among r-colorable graphs the Turán graph is the unique graph, which has the most number of edges. Proof. Local change. 5

6 Turán s Theorem The Turán number exn, H of a graph H is the largest integer m such that there exists an H-free graph on n vertices with m edges. Example: Mantel s Theorem states exn, K 3 = n 4. Theorem. Turán, 1941 exn, K r = et n,r 1 = 1 1 r 1 n +On. Proof. Prove by induction on r that G K r = there is an r 1-partite graph H with V H = V G and eh eg. Then apply the Lemma to finish the proof. Here H-free means that there is no subgraph isomorphic to H 6

7 Turán-type problems Question. Turán, 1941 What happens if instead of K 4, which is the graph of the tetrahedron, we forbid the graph of some other platonic polyhedra? How many edges can a graph without an octahedron or cube, or dodecahedron or icosahedron have? The platonic solids 7

8 Erdős-Simonovits-Stone Theorem Theorem. Erdős-Stone, 1946 For arbitrary fixed integers r and t 1 exn, T rt,r = 1 1 n + on. r 1 Corollary. Erdős-Simonovits, 1966 For any graph H, 1 exn, H = 1 χh 1 n + on. Corollaries of the Corollary. exn, octahedron = n 4 + on exn, dodecahedron = n 4 + on exn, icosahedron = n 3 + on exn, cube = on 8

9 Proof of the Erdős-Simonovits Corollary Theorem. Erdős-Stone, 1946 For arbitrary fixed integers r and t 1 exn, T rt,r = 1 1 n + on. r 1 Corollary. Erdős-Simonovits, 1966 For any graph H, 1 exn, H = 1 χh 1 n + on. Proof of the Corollary. Let r = χh. χt n,r 1 < χh, so et n,r 1 exn, H. T rα,r H, so exn, T rα,r exn, H, where α is a constant depending on H; say α = αh. 9

10 The number of edges in a C 4 -free graph Theorem Erdős, 1938 exn, C 4 = On 3/ Proof. Let G be a C 4 -free graph on n vertices. C = CG := number of K 1, cherries in G. Doublecount C. Counting by the midpoint: Every vertex v is the midpoint of exactly dv cherries. Hence C = v V dv. Counting by the endpoints: Every pair {u, w} of vertices form the endpoints of at most one cherry. Otherwise there is a C 4 G. Hence C 1 n. 10

11 Proof cont d Combine and apply Jensen s inequality Note that x x is a convex function n dg = 1 n C v V dv dg n v V dv is the average degree of G. n 1 dg dg 1. Hence n dg. Theorem E. Klein, 1938 exn, C 4 = Θn 3/ Proof. Homework. Theorem Kővári-Sós-Turán, 1954 For s t 1 Proof. Homework. exn, K t,s c s n 1 t 11

12 Open problems and Conjectures Known results. Ωn 3/ exn, Q 3 On 8/5 Ωn 9/8 exn, C 8 On 5/4 Ωn 5/3 exn, K 4,4 On 7/4 Conjectures. exn, K t,s = Θ exn, C k = Θ exn, Q 3 = Θ n 1 min{t,s} n 1+1 k n 8 5 If H is a d-degenerate bipartite graph, then exn, H = O n 1 d true for t =, 3 and s t or t 4 and s > t 1! true for k =, 3 and 5. 1

13 Proof of the Erdős-Stone Thm Erdős-Stone Theorem. Understanding precisely what it actually says For any ɛ > 0 and integers r, t 1 there exists an integer M = Mr, t, ɛ, such that any graph G on n M vertices with more than 1 1 r 1 + ɛ n edges contains Trt,r. We derive this through the following statement. Seemingly Weaker Theorem. For any ɛ > 0 and integers r, t 1 there exists an integer N = Nr, t, ɛ, such that any graph G on n N vertices and with δg 1 1 r 1 + ɛ n contains T rt,r. Note that w.l.o.g. ɛ < 1 r 1. 13

14 Derivation of the Erdős-Stone Theorem from the Seemingly Weaker Theorem. Let G be a graph on n Mr, t, ɛ vertices with more than 1 r ɛ n edges. Recursively delete vertices which are adjacent to less than 1 r ɛ fraction of the remaining vertices. What is the number n of vertices we are left with? We deleted at most edges. So eg This implies n j=n +1 j 1 1 r 1 + ɛ n + 1 n ɛ n n 1 r 1 ɛ n r 1 + ɛ n. - + n We choose Mr, t, ɛ such that n Mr, t, ɛ implies n Nr, t, ɛ/. At this point we don t know Mr, t, ɛ yet!!! We ll define it in the proof through Nr, t, ɛ/. which is known! 14.

15 Proof of the Seemingly Weaker Theorem. Induction on r. For r = the claim is true provided ɛn t n n t > t 1, which is certainly true from some threshold N, t, ɛ. Let r and G be a graph on n Nr + 1, t, ɛ vertices with δg 1 1 r + ɛ n. We would like to find a T r+1t,r+1 in G. Let s = t ɛ. By the induction hypothesis there is a T rs,r in G with vertex-set A 1... A r, where A 1 =... = A r = s. U = V G \ A 1... A r. W = {w U : Nw A i t, i = 1,..., r} is the set of vertices eligible to extend some part of A 1,..., A r into a T r+1t,r+1. Again, we don t know Nr + 1, t, ɛ yet. Here we assume Nr + 1, t, ɛ Nr, s, ɛ. 15

16 Double-count the number of edges missing between U and A 1... A r. They are at least U W s t s tn if W is small at most rs 1 r ɛ n s rtn, if W is small From this we have W r 1ɛ 1 ɛ n rs Thus if n is large enough then W > s rt 1. t So we can select t vertices from W, which are adjacent to the same t vertices in each A i. If Nr + 1, t, ɛ > s rt t 1 + rs 1 ɛ r 1ɛ 16