1 Cliques and Independent Sets. Definition 1

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1 1 Cliques and Independent Sets Definition 1 A set of vertices is called independent if no two vertices in the set are adjacent. A set of vertices is called a clique if every two vertices in the set are adjacent. An independent set (resp. a clique) is called maximal, if no other independent set(resp. a clique) contains it. An independent set (resp. a clique) is called maximum, if its cardinality is maximal among all independent sets (resp. cliques) in the graph. α(g) (resp. ω(g)) denotes the maximum size of an independent set (resp. a clique) in a graph G; π(g) denotes the maximum size of a matching in a graph G; 1

2 Find a maximum independent set and a maximum clique Problem: Given n 1 and p (1 p n), construct a graph with n vertices and clique-size p, which contains the maximum number of edges. Dual Problem: Given n 1 and p (1 p n), construct an n-vertex graph with the minimal number of edges for which the size of any independent set is p. Special Case: (p = 2) Find the maximum number of edges in a graph with n vertices and without triangles. 2

3 Theorem 1 (a) Every triangle-free graph with 2k vertices has at most k 2 edges. (b) The only triangle-free graph with 2k vertices and k 2 edges is the complete bipartite graph with the partitions of sizes k each. Proof. Induction on k. PART (a). Base. The result is straightforward for k = 1. Inductive step. Let the statement be correct for all triangle-free graphs with 2k 2 vertices, and let G be a triangle-free graph with 2k vertices. Select an edge (a,b) E(G). The set of all edges of G consists of (i) the edges in G {a,b}; (ii) the edges connecting {a,b} with the rest of G; and (iii) the edge (a,b) itself. By induction, E(G {a,b}) (k 1) 2. Furthermore, there are at most 2k 2 edges of type (ii). (Explain why) a b 2k 2 vertices; no triangles e(g) e(g {a,b})+2k 2+1 (k 1) 2 +2k 1 = k 2 2k +1+2k 1 = k 2. 3

4 PART (b). DenoteT(2k,2)thecompletebipartitegraphon2k verticeswiththe partitions of sizes k each. It is easy to see that for k = 1, the only triangle-free graph with k 2 edges is T(2k,2). Assume, inductively, that the second part of the Theorem holds for all graphs with 2k 2 vertices, and let G be a 2k-vertex triangle-free graph with k 2 edges. Then, as before consider G {a,b} for some edge ab and analyze the inequality e(g) e(g {a,b})+2k 2+1 (k 1) 2 +2k 1 = k 2. The following is obvious: for this inequality to be an equality, G {a,b} must have exactly (k 1) 2 edges and the number of edges connecting {a,b} with the rest must be equal to 2k 2. The first requirement implies, inductively, that G {a,b} = T(2k 2,2), which in turn, implies that a (resp. b) is adjacent to the vertices of one part of G {a,b} only. Those parts must be distinct which proves that G = T(2k,2). 4

5 Comment. The Theorem above can be expanded to graphs with an arbitrary number n of vertices (even or odd): Theorem 2 (a) Every triangle-free graph with n vertices has at most n2 4 edges. (b) The only triangle-free graph with n vertices and n2 4 edges is the complete bipartite graph with the partitions of sizes n 2 and n

6 Definition 2 Givennandp,Turán sgrapht(n,p)obtainedbypartitioningnverticesintopdisjointsetsofalmostthesamesize(within1)andsetting edges to be all pairs comprised of vertices from different partitions. The sizes of the partitions of T(n,p) are obtained by dividing n by p with a remainder: n = p q +r, where 0 r p 1. The r partitions of T(n,p) are of size q+1, and the remaining p r partitions are of size q. It is easy to prove the following Lemma 1 If n p, then T(n,p) is a complete graph on n vertices. For any n 2 and 1 p < n e(t(n+p,p)) = p +n(p 1)+e(T(n,p)). 2 Theorem 3 (Turán[1944]) Given positive integers n and p, the number of edges of any graph with n vertices and without a clique of size p+1 is at most e(t(n,p)). 6

7 Theorem 4 (Turán[1944]) Every graph of order n and size m contains an independent set of size n 2 /(2m+n). Proof. We present a greedy algorithm which constructs an independent set whose size is n 2 /(2m+n). I = ; H = G; while (not done) select a vertex v V(H) of the minimal degree in H; I = I {v}; H = H {v} {all the vertices adjacent to v}; To prove that the algorithm constructs an independent set of size n 2 /(2m+n), we use induction on n. The statement is obviously true for the 1-vertex graph. Suppose, the theorem holds for every graph with < n vertices and let G be a graph with n vertices and m edges. Let d be the degree of the vertex v chosen by the algorithm. Consider the graph H resulting from the deletion of v and all vertices adjacent to v. Clearly, the degree of every deleted vertex is at least d. Therefore, thetotalnumberofedgesdeletedisatleastd(d+1)/2. Thus, m(h) m d(d+1)/2 and n(h) = n d 1. Since the algorithm constructs an independent set in H and adjoins v to it, the theorem will be proved if we verify the following inequality 1+ (n d 1) 2 2(m d(d+1)/2)+n d 1 n2 2m+n 7

8 The left part can be transformed as follows: 1+ (n d 1) 2 2(m d(d+1)/2)+n d 1 = 1+ (n (d+1))2 2m (d+1) 2 +n = 2m+n+n2 2n(d+1) 2m+n (d+1) 2 Let us now denote 2m+n by Q. We must show that Indeed Q+n 2 2n(d+1) Q (d+1) 2 n2 Q. Q 2 +Qn 2 2nQ(d+1) Qn 2 n 2 (d+1) 2 (Q n(d+1))

9 Problem 1 Determine the disconnected n-vertex graphs (n 2) that have the maximum number of edges. Problem 2 Determine the maximum number of edges in an n- vertex graph (without parallel edges) that has an independent set of size α. Problem 3 Let G be a simple graph with n 4 vertices. Prove that if G has more than n 2 /4 edges, then it has a vertex whose deletion leaves a graph with more than (n 1) 2 /4 edges. Problem 4 Prove that every n-vertex triangle-free simple graph with the maximum number of edges is isomorphic to K n/2, n/2. Problem 5 A flat circular city of radius six miles is patrolled by eighteen police cars, which communicate with one another by radio. If the range of a radio is nine miles, show that at any time, there is always at least two cars each of which can communicate with at least five other cars. 9

10 2 Dominating sets Definition 3 For a graph G, a set D V(G) of vertices is called dominating if N G (D) = V(G), that is if every vertex in V(G) is either in D or adjacent to a veretex in D. A dominating set is called minimal if no subsets of D is dominating. A dominating set is called minimum if no smaller set in G is dominating. γ(g) denotes the minumal size of a dominating set in a graph G. Lower and upper bounds for γ(g). Theorem. For a simple graph G, letα(g) denote the maximal size of an independent set, and let diam(g) denote the diameter of G. Then diam(g)+1 γ(g) α(g). 3 10