Turán numbers of vertex-disjoint cliques in r-partite graphs

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1 Turán numbers of vertex-disjoint cliques in r-partite graphs Jessica De Silva a,1,, Kristin Heysse b, Adam Kapilow c,, Anna Schenfisch d,, Michael Young e, a University of Nebraska-Lincoln, United States b Macalester College, Saint Paul, MN, 55105, United States c University of Washington, United States d Montana State University, United States e Iowa State University, United States Abstract For two graphs G and H, the Turán number ex(g, H) is the maximum number of edges in a subgraph of G that contains no copy of H. Chen, Li, and Tu determined the Turán numbers ex(k m,n, kk ) for all k 1 [5]. In this paper we will determine the Turán numbers ex(k a1,...,a r, kk r ) for all r 3 and k Introduction All graphs considered here are finite, undirected, and simple. Throughout the paper we use the standard graph theory notation (see [3]). In particular, a graph is called a complete r-partite graph if its vertex set can be partitioned into r independent sets V 1,..., V r such that for any i = 1,,..., r every vertex in V i is adjacent to all other vertices in V j, j i. We denote a complete r-partite graph with part sizes V i = n i by K n1,...,n r. For a graph G and a positive integer k we use kg to denote k vertex-disjoint copies of G. Given S V (G), the subgraph of G induced by S will be denoted G[S] and the subgraph G[V (G)\S] will be denoted G\S. For two graphs G and H, G + H is the join of G and H, that is Principal corresponding author addresses: jessica.desilva@huskers.unl.edu (Jessica De Silva), kheysse@macalester.edu (Kristin Heysse), akapilow@uw.edu (Adam Kapilow), annaschenfisch@montana.edu (Anna Schenfisch), myoung@iastate.edu (Michael Young) 1 Research is supported in part by the NSF-GRFP Grant DGE Research was supported in part by the NSA Grant H

2 the graph obtained from G H by adding every edge containing a vertex of G and a vertex of H. For two graphs G and H, the Turán number, or extremal number, ex(g, H) is the maximum number of edges among all H free subgraphs of a host graph G. The study of such numbers began in 1907 when Willem Mantel determined the maximum number of edges in a triangle-free graph on n vertices, i.e. ex(k n, K 3 ). In 1941, this theorem was strengthened by Pál Turán who determined ex(k n, K r ). Since then, the most well-studied host graphs have been the complete graph K n and the complete bipartite graph K m,n. Recent studies of extremal numbers consider the case when the forbidden graph H is made up of several vertex-disjoint copies of some smaller graph (e.g., [4], [9], [7], [11]). In particular, Chen, Li, and Tu determined ex(k m,n, kk ) = m(k 1) for 1 k n m [5]. The focus of this paper is to extend their result to forbidding vertex-disjoint cliques of size r in a complete r-partite graph. Theorem 1 (Main Theorem). For any integers 1 k n 1... n r, ex(k n1,...,n r, kk r ) = n i n j n 1 n + n (k 1). For the lower bound, clearly ((n 1 k + 1)K 1 K k 1,n ) + K n3,...,n r is such a subgraph of K n1,...,n r with the required number of edges and no copy of kk r. This gives the lower bound and the remainder of this paper will work to establish the upper bound. The upper bound is proven by considering two cases: n = n r and n < n r. In the former case, the proof is inductive on n 1 + k with Lemmas and 3 as the base cases. In the latter case, the proof is inductive on the total number of vertices in the host graph.. Main results The proof of Theorem 1 first considers the case when the host graph is almost balanced, that is every part size is the same except the smallest part. This proof requires a double induction and is preceded by the two necessary base cases.

3 Define h k (n 1, n,..., n r ) = n i n j n 1 n + n (k 1). Given two disjoint subsets of the vertex set, A, B V (G), define AB as the graph formed by the set of edges in G incident to a vertex in A and a vertex in B. For ease of notation, given an r-partite graph G with parts V 1,..., V r we let R(G, r) = {{v 1,..., v r } V (G) : v i V i for all i [r]}, that is R(G, r) is the set of all r-tuples of vertices with exactly one vertex from each part. We utilize R(G, r) throughout to facilitate the counting of edges. For S R(G, r) define w(s) = E(G[S]). Note that for S R(G, r) an edge v i v j V i V j is counted in w(s) if and only if both v i and v j are present in S, therefore, summing over all S R(G, r), w(s) = E(V i V j ) n l. (1) Lemma. For 1 n 1 n, l i,j Proof. Suppose G K n1,n,...,n ex(k n1,n,...,n, K r ) = h 1 (n 1, n,..., n ). S R(G, r), w(s) ( r ) 1 and hence w(s) does not contain a copy of K r. Then for all (( ) ) r 1 n 1 n r 1. () 3

4 Subtracting () from (1) yields, 0 r j= E(V 1 V j ) n r + = n 1 n r 3 E(G) + r j= E(V i V j ) n 1 n r 3 E(V 1 V j ) n r 3 (n n 1 ) ( ( ) ) r 1 n 1 n r 3 E(G) + E(G) n ( ) r 1 = n r E(G) n r 1 (n n 1 ) ( r 1 = n r E(G) (r )n 1 n r 1 n r 3 (( r ) n r. (( ) r (( ) r ) 1 n 1 n r 1 (n n 1 ) ) ) 1 n 1 n r 1 ) 1 n 1 n r 1 (( ) ) r 1 n 1 n r 1 Therefore ( ) r 1 E(G) n 1 n (r 1) + n n 1 n = h 1 (n 1, n,..., n ). Lemma 3. For 1 n 1 n, ex(k n1,n,...,n, n 1 K r ) = h n1 (n 1, n,..., n ). Proof. This proof is by induction on n 1. The base case of n 1 = 1 is true for all positive integers n by Lemma. Assume the statement holds for n 1 < n 1 where n 1. Suppose G K n1,n,...,n contains more than h n1 (n 1, n,..., n ) edges and does not contain a copy of n 1 K r. We have E(G) > h n1 (n 1, n,..., n ) > h 1 (n 1, n,..., n ), which implies G contains a copy of K r. Let S R(G, r) such that G[S] = K r. Then E(G\S) h n1 1(n 1 1, n 1,..., n 1), otherwise G\S contains a copy of (n 1 1)K r, and this together with G[S] is a copy of n 1 K r in G. Therefore, E(G) E(G\S) > h n1 (n 1, n,..., n ) h n1 1(n 1 1, n 1,..., n 1) ( ) r = (r 1)(n 1 + (r )n ) + (r 1)n 1. 4

5 Since the number of edges in K n1,n,...,n with a vertex in S is ( ) r (r 1)(n 1 + (r )n ) + (r 1)n, this implies all edges in the host graph containing a vertex in S are present in G. Note that this holds for every S such that G[S] = K r in G. Let u i V i and u j V j with i j, if neither u i nor u j is in S, then u i u j E(G). Otherwise, for v i S V i, let S = (S\{v i }) {u i }. S induces a copy of K r in G and therefore u i u j E(G). Hence G = K n1,n,...,n contains n 1 K r which is a contradiction. We now prove Theorem 1. and thus The proof considers two cases: n = n r and n < n r. The first case is proven using double induction and relies on Lemmas and 3 as the base cases. The second case is proven by induction on the total number of vertices in the host graph. Proof. Let 1 k n 1 n r. Case 1. Assume n = n r, we proceed by induction on n 1 + k. The base case of k = 1 is true for all positive integers n 1 by Lemma and the base case of n 1 = k is true for all positive integers k n 1 by Lemma 3. Assume the statement is true for parameters n 1, k such that n 1 + k < n 1 + k where n 1 > k, and that G K n1,n,...,n does not contain a copy of kk r. We first obtain a lower bound on the number of (not necessarily disjoint) copies of K r in G. Suppose there are exactly q such copies of K r in G, then ( ) (( ) ) r r w(s) q + (n 1 n r 1 q) 1. Recall that this gives, q r j= w(s) = r j= E(V 1 V j ) n r + E(V 1 V j ) n r + E(V i V j ) n 1 n r 3 n 1 n r 1 E(V i V j ) n 1 n r 3, (( ) ) r 1. (3) 5

6 We will use (3) to get an upper bound on E(G) by counting E(G\S). An edge v i v j V i V j is counted in E(G \ S) if and only if v i S and v j S, hence r E(G\S) = E(V 1 V j ) (n 1 1)(n 1)n r + E(V i V j ) (n 1) n 1 n r 3. (4) j= Using (3) and (4), ( E(G\S) + q + n 1 n r 1 (( r )) 1 (n 1) ) r j= + Now for S R(G, r), suppose G[S] is a copy of K r. E(V 1 V j ) n 1 (n 1)n r E(V i V j ) (n 1)n 1 n r = E(G) (n 1)n r n 1. (5) Then E(G\S) h k 1 (n 1 1, n 1,..., n 1) else by induction G\S contains a copy of (k 1)K r and so this together with G[S] yields a copy of kk r in G. If G[S] is not complete, then since G\S does not contain a copy of kk r induction gives E(G\S) h k (n 1 1, n 1,..., n 1). Hence E(G \ S) q ( h k 1 (n 1 1, n 1,..., n 1) ) + (n 1 n r 1 q) ( h k (n 1 1, n 1,..., n 1) ) = q(1 n ) + n 1 n r 1 (h k (n 1 1, n 1,..., n 1)) and thus, using (5), we have ( (( E(G) (n 1)n r n 1 n 1 n r 1 r ) ) h k (n 1 1, n 1,..., n 1) + 1 (n 1). ) Therefore E(G) n n 1 ( h k (n 1 1, n 1,..., n 1) + = h k (n 1, n, n,..., n ). (( ) ) ) r 1 (n 1) Case. Assume n < n r, we proceed by induction on the number of total vertices. The base case of n 1 = n r is true for all positive integers k by Case 1. Assume the statement holds for all parameters n 1,..., n r such that r i=1 n i < 6

7 r i=1 n i. Suppose G K n1,...,n r does not contain a copy of kk r. Let v r V r, the graph G\{v r } does not contain a copy of kk r, has fewer vertices than G, and n n r 1. Therefore E(G) = E(G \ {v r }) + d(v r ) ex(k n1,...,n r 1, kk r ) + d(v r ) = h k (n 1,..., n r 1) + d(v r ) = r 1 n i n j n i n 1 n + n (k 1) + d(v r ) i=1 = h k (n 1, n,..., n r ). n i n j n 1 n + n (k 1) 3. Concluding remarks The main theorem relies on the fact that both K r and K n1,n,...,n r are r- partite. Certainly the host graph must be l-partite for l r to have K r as a subgraph. An interesting generalization would be to calculate ex(k n1,n,...,n l, kk r ) for r < l. In [6], De Silva, Heysse, and Young proved that ( l ) ex(k n1,n,...,n l, kk ) = (k 1) n i, i= however the Turán number is open for r 3. The graph ((n 1 + n k + 1)K 1 K k 1,n3 ) + n 4 K 1 does not contain kk 3, hence ex(k n1,n,n 3,n 4, kk 3 ) (n 1 + n + n 3 )n 4 + (k 1)n 3. This construction can be easily generalized to r-partite graphs, but it is not clear that this is an extremal construction. 7

8 We also note that many of the results cited in this paper were originally considered in conjunction with the rainbow number rb(g, H). For a graph G and subgraph H, rb(g, H) is the minimum number of colors required to ensure that every edge coloring of G with rb(g, H) colors has a rainbow copy of H (where a subgraph is rainbow if it has no two edges with the same color). Often the rainbow number is proven via the analogous Turán number, and it would be interesting to see this work extended to the rainbow number rb(k n1,...,n r, kk r ). 4. References [1] H. Bielak, S. Kieliszek, The Turán number of the graph 3P 4, Annales Universitatis Mariae Curie-Sklodowska, sectio A Mathematica 68 (1) (014) 1 9. [] H. Bielak, S. Kieliszek, The Turán number of the graph P 5, Discuss. Math. Graph Theory 36 (3) (016) [3] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, Macmillan, London, 1976, Elsevier, New York. [4] N. Bushaw, N. Kettle, Turán numbers of multiple paths and equibipartite forests, Combinatorics, Probability and Computing, 0 (011) [5] H. Chen, X. Li, J. Tu, Complete solution for the rainbow numbers of matchings, Discrete Math. 309 (10) (009) [6] J. De Silva, K. Heysse, M. Young, Rainbow number for matchings in r- partite graphs, preprint. [7] I. Gorgol, Turán numbers for disjoint copies of graphs, Graphs and Comb. 7 (5) (011) [8] X. Li, J. Tu, Z. Jin, Bipartite rainbow numbers of matchings, Discrete Math. 309 (009)

9 [9] B. Lidický, H. Liu, C. Palmer, On the Turán number of forests, Electr. J. Comb. 0 () (01) #P6. [10] L-T. Yuan, X-D. Zhang, Turán numbers for disjoint paths, arxiv: [11] L-T. Yuan, X-D. Zhang, The Turán number of disjoint copies of paths, Discrete Math. 340 () (017)