Ma/CS 6b Class 19: Extremal Graph Theory

Transcription

1 /9/05 Ma/CS 6b Class 9: Extremal Graph Theory Paul Turá By Adam Sheffer Extremal Graph Theory The subfield of extremal graph theory deals with questios of the form: What is the maximum umber of edges that a graph with vertices ca have without cotaiig a give subgraph H? Characterize the graphs that obtai the this maximum umber of edges A graph without K 5

2 /9/05 ex, H We deote by ex, H the maximum umber of edges i a graph with vertices ad o subgraph H Example Recall that P i is a simple path of legth i What is ex 4, P 3? 3 Graphs Without Triagles Problem Fid ex, K 3 What graph cotais a large umber of edges but does ot have K 3 as a subgraph? Ay complete bipartite graph

3 /9/05 Maximizig the Number of Edges What value of m maximizes the umber of edges i K m, m? The umber of edges is m m The maximum of /4 is obtaied for G /,/ Ca we fid a graph without triagles ad with more edges? Matel s Theorem Theorem ex, K 3 = /4 Proof Let G = V, E be a triagle-free graph with V = d i = deg v i If v i, v j E, the v i ad v j do ot have commo eighbors, so d i + d j We thus have d i + d j v i,v j E E Sice every d i appears i d i elemets E d i + d j d i,d j E = d i i 3

4 /9/05 The Cauchy Schwarz Iequality The Cauchy Schwarz iequality For ay a, a,, a, b, b,, b R, we have a i b i a i Equality holds iff the vectors a,, a ad b,, b are liearly depedet Available olie b i Completig the Proof We proved that Recall that E d i + d j d i,d j E i d i = E = d i By Cauchy-Schwarz with a i = d i ad b i = : d i We thus have E d i i i d i i 4 E i i E 4 4

5 /9/05 The Maximum Graph is Uique Claim The oly graph that has the maximum umber of edges ex, K 3 = /4 is K /, / Proof We cosider the case of a eve I our proof for ex, K 3 = /4, we used Cauchy-Schwarz to obtai d i i i d i i, which implied E i d i 4 E Equality holds if ad oly if for every i, we have d i = / Iequality of Arithmetic ad Geometric Meas AM-GM Iequality For ay a, a,, a R, we have a + + a a a / No Shirt available Marketig opportuity? 5

6 /9/05 Recall: Idepedet Sets Cosider a graph G = V, E A idepedet set i G is a subset V V such that there is o edge betwee ay two vertices of V Fidig a maximum idepedet set i a graph is a major problem i theoretical computer sciece No polyomial-time algorithm is kow Matel s Theorem: Secod Proof G = V, E a triagle-free graph with V = α the size of the largest idepedet set A i G The set of eighbors of ay v V is a idepedet set, so α deg v B = V A, so B = α Also, B is a vertex cover By the AM-GM iequality, we have α + α E deg v α B = 4 v B 6

7 /9/05 Geeralizig the Problem We move from ex, K 3 to ex, K r For some r > 3, what graph cotais may edges but o K r? A r-partite graph is a graph with r parts, ad o edge betwee two vertices of the same part (geeralizig bipartite graphs) Example The figure presets the complete 4-partite graph K 3,3,3,4 Turá Graphs The Turá graph T, r is a complete r-partite graph with vertices, such that each part cosists of either /r or /r vertices The graph i the figure is T 3,4 T, r cotais o copy of K r ad has about r / edges 7

8 /9/05 Turá s Theorem Theorem If G = V, E is a graph that cotais o copy of K r ad V =, the E r Proof By iductio o Iductio basis Easily holds whe < r Iductio step G = V, E a graph that cotais o copy of K r, with V = ad E = ex, K r G cotais a copy of K r Otherwise, addig a edge to G would ot yield a copy of K r, cotradictig the maximality of G G = V, E a graph that cotais o copy of K r, with V = ad E = ex, K r edges A a iduced subgraph of G that is a K r r A cotais edges By the hypothesis, G A has at most ( r+) edges r Sice each vertex of V A is adjacet to at most r vertices of A, there are r + r edges betwee V\A ad A Combiig the above, we have r r r + E + / r + r + r / r 8

9 /9/05 Which Actor from the Big Bag Theory is a Real Scietist? Proof #: Balacig Weights G = V, E a graph that cotais o copy of K r, with V = v,, v We assig a weight w i = to every v i We set W = v w i w i,v j E j Iitially we have W = E weights to icrease W /3 We wish to move / /3 / = 9 / = 4 9

10 /9/05 Balacig Weights (cot) v i, v j two vertices of V with positive weights, such that v i, v j E N i, N j the sum of the weights of the eighbors of v i ad v j, respectively WLOG, assume that N i N j By movig the weight of v i to v j, we chage W by w i N j w i N i v i v i vj 0 0 vj 0 More Balacig of Weights If there exist v i, v j with positive weights ad v i, v j E, we ca move the weight vertex of oe to the other without decreasig W We repeatedly do this util the etire weight is o the vertices of a complete subgraph K m Cosider v i, v j K m with w j w i ε > 0 The movig ε from w j to w i chages W by ε w i ε w j = ε w j w i ε ε

11 /9/05 Cocludig the Proof We coclude that we ca move the etire weight to a subgraph K m where each vertex of the subgraph has a weight of m By the assumptio, m r That is, we obtai W r Recall that we started with W = E Sice we oly icreased W, we have E r r = r (r ) r E r Cliques Give a graph G = V, E, a clique of G is ay subgraph that is a complete graph No polyomial-time algorithm for fidig a maximum clique i a graph is kow The problem is equivalet to fidig a maximum idepedet set i a graph Defie the graph G c = (V, E c ) such that v, u E c iff v, u E Fidig a maximum clique i G is equivalet to fidig a maximum idepedet set i G c

12 /9/05 A Lower Boud o the Clique Size Lemma Let G = V, E be a graph with V = v,, v, ad let d i = deg v i The G cotais a clique of size at least d i = Probabilistic Proof We uiformly choose a permutatio π of,, We build a subset C π V: for every i, we add v π i to C π if v π i is adjacet to v π, v π,, v π i Notice that C π is a clique X i a idicator variable for whether v π i C π E X i = Pr X i = = d π i

13 /9/05 Proof (cot) X i a idicator variable for whether v π i C π E X i = Pr X i = = d π i Set X = X i X is the size of the clique C π By liearity of expectatio E X = E X i = d π i = d i Completig the Proof X is the size of the clique C π E X = d i Thus, there exists a permutatio that leads to a clique of at least this size 3

14 /9/05 Proof #3 of Turá s Theorem G = V, E a graph that cotais o copy of K r, with V = v,, v d i = deg v i, a i = d i, b i = d i By Cauchy-Schwarz a i b i a i Sice a i b i =, we have a i b i b i d i = deg v i, a i = d i, b i = We have a i b i = d i d i d i By the previous lemma, G has a clique of size at least d i By the assumptio, r d i d i r Thus, = r E r E E r 4

15 /9/05 Recap We saw three proofs of Turá s Theorem: A simple proof by iductio A proof by assigig a weight to every vertex ad maipulatig these weights A proof by combiig the probabilistic method ad the Cauchy-Schwarz iequality The Ed: Mayim Bialik Just like the character Amy Farrah Fowler that she plays i the show, Mayim Bialik has a PhD i eurosciece From UCLA 5