Turán's Theorem and Maximal Degrees

Journal of Combinatorial Theory, Series B 75, 160164 (1999) Article ID jctb.1998.1873, available online at http:www.idealibrary.com on

NOTE Turan's Theorem and Maximal Degrees Bela Bollobas Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152, and Trinity College, Cambridge CB2 1TQ, England Received September 18, 1997

Extending results of Bollobas and Thomason (1981, J. Combin. Theory Ser. B 31, 111114) and Bondy (1983, J. Combin. Theory Ser. B 34, 109111), we characterize graphs of order n and size at least t r(n) that do not have a vertex x of maximal degree d x whose neighbours span at least t r&1(d x )+1 edges. Furthermore, we show that, for every graph G of order n and size at least t r(n), the degree-greedy algorithm used by Bondy (1983, J. Combin. Theory Ser. B 34, 109111) and Bollobas and Thomason (1985, Ann. Discr. Math. 28, 4797) constructs a complete graph K r+1 , unless G is the Turan graph T r(n).  1999 Academic Press

For every r1 and n1 there is a unique r-partite graph of order n with maximal size, the Turan graph T r(n). This is the complete r-partite graph with n vertices and as equal classes as possible. In particular, if n=rs then T r(n) is just K r(s), the complete r-partite graph with s vertices in each class. The classical theorem of Turan [6] states that if a graph G of order n has at least t r(n) edges, where t r(n)=e(T r(n)), then either G is exactly T r(n), or else it contains a K r+1 , a complete graph of order r+1. Bollobas and Thomason [1] proved the stronger result, conjectured by Erdo s, that if G is a graph of order n and size at least t r(n) then either G=T r(n) or else there is a vertex x such that G x =G[1(x)], the subgraph spanned by the neighbours of x, contains at least t r&1(d x )+1 edges, where d x is the degree of x. Furthermore, d x is not much smaller than the maximal degree of T r(n). By making use of an earlier proof of Turan's theorem by Erdo s [5] (see also [7]), Bondy [3] showed that if G has more than t r(n) edges then every vertex of maximal degree will do for x. Theorem 1. If a graph G of order n has more than t r(n) edges and x is a vertex of maximal degree d x , then e(G x )=e(G[1(x)])t r&1(d x )+1. In searching for an extension of Theorem 1 to the case e(G)t r(n), Bondy [3] (corrected in [4]) noted that, ``somewhat surprisingly,... the 160 0095-895699 30.00 Copyright  1999 by Academic Press All rights of reproduction in any form reserved.

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following slight modification of the statement of [Theorem 1] is false: Let G be a simple graph on n vertices and at least t r(n) edges, and let x be a vertex in G of degree d x =2(G). Then either G=T r(n) or the subgraph induced by the neighbours of x has more than t r&1(d x ) edges.'' The main results of this note, Theorems 2 and 3, show that, in spite of this phenomenon, Theorem 1 does have natural extensions that cover the case e(G)t r(n) as well. The following simple algorithm was used in [2] and [3] to construct a large clique in a graph G. Pick a vertex x 1 of maximal degree in G 1 =G, then a vertex x 2 of maximal degree in G 2 =G x1 =G 1[1(x 1 )], the subgraph of G 1 induced by the neighbours of x 1 in G 1 , and so on. The algorithm stops with x l if x l has no neighbours in G l . The output is the clique with vertex set [x 1 , ..., x l ]. In [2] this algorithm was called the hammer algorithm; here we shall call it the degree-greedy algorithm. In [3] Bondy also noted the immediate consequence of Theorem 1 that the degree-greedy algorithm always constructs a complete subgraph of order r+1 in every graph of order n and size at least t r(n)+1. As we shall see, Theorem 2 implies that the degree-greedy algorithm always constructs a complete graph of order at least r+1 in every graph G{T r(n) with n vertices and at least t r(n) edges. Before stating our results, let us list some simple properties of the Turan graph T r(n). Set $ r(n)=n&WnrX and 2 r(n)=n&wnrx . Clearly, the minimal degree of T r(n) is $(T r(n))=$ r(n) and its maximal degree is 2(T r(n)) =2 r(n), so the degrees are as equal as possible, given that their sum is 2t r(n). In particular, if G is a graph of order n and size t r(n), and 2(G) 2 r(n), then G has at least as many vertices of degree 2 r(n) as T r(n). Also, if x # T r(n) is a vertex of minimal degree then the graph T r(n)&x, obtained from T r(n) by deleting x, is precisely T r(n&1). If H is a graph of order n&kt r&1(n&k) unless k is wnrx or WnrX , in which case we have equality. From these observations it is but a short step to the following extension of Theorem 1. Theorem 2. Let G be a graph with n vertices and t r(n)+a edges, where a0. Let x be a vertex of maximal degree d x =2(G)=n&k. Set W=1(x), U=V(G)"W and G x =G[W]. Then e(G x )t r&1(d x )+a, and the inequality is strict unless k=wnrx , d x =2 r(n), and U is a set of independent vertices, each of degree d x =n&k, so that G=G x +K k .

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Proof. Since n&k=2(G)2(T r(n))W2e(G)nX2 r(n)=n&wnrx , we have kwnrx . Assume that e(G x )t r&1(n&k)+a. Writing e(U) for e(G[U]), and e(U, W) for the number of edges from U to W, we find that e(G)=e(G x )+e(U)+e(U, W)=e(G x )+(e(U)+ 12 e(U, W))+ 12 e(U, W) =e(G x )+ 12 : d u + 12 e(U, W) u#U

e(G x )+k(n&k)t r&1(n&k)+a+k(n&k)t r(n)+a. Consequently, k=wnrx , d x =2 r(n), e(G x )=t r&1(d x )+a and e(U, W)= k(n&k), so that every vertex in U has degree n&k, and G=G x +K k , as claimed. K By repeated applications of Theorem 2 we can get detailed information about graphs G of order n and size t r(n) that do not have a vertex x of maximal degree with e(G x )t r&1(d x )+1. Theorem 3. Let G{T r(n) be a graph of order n=rs+ p and size at least t r(n), where r2, s1 and 0 pt r(n) in Theorem 1 to e(G)t r(n).

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Corollary 4. Let G{T r(n) be a graph of order n=rs+ p and size tr(n), where s1 and 0 pt r&1(d x ), then we are done by the induction hypothesis since the degree-greedy algorithm finds at least r vertices in G x : these vertices and x span a complete graph of order at least r+1. (This is also the case covered in [3].) Otherwise, by Theorem 2, d x =2 r(n)=n&k, where k=wnrx , e(G x )=t r&1(n&k) and G=G x +K k . Hence, by another application of the induction hypothesis to G x , we see that either lr+1 or else G x =T r&2(n&k) and G=T r&1(n), as claimed. K In conclusion we note that the natural strengthening of the original conjecture of Erdo s mentioned in the introduction is still open. Given nr1, let d r(n) be the maximal integer such that every graph G{T r(n) of order n and size t r(n) has a vertex x of degree d x d r(n) with e(G x ) tr&1(d x )+1. The conjecture of Erdo s was that d r(n) exists for every pair (r, n) and is not very large: this was proved in [1]. Also, we know from Corollary 4 that d r(n)=2(n)=n&wnrx if n#0 or 1 (mod r). However, it would be desirable to determine d r(n) for all pairs (r, n).

REFERENCES 1. B. Bollobas and A. Thomason, Dense neighbourhoods and Turan's theorem, J. Combin. Theory Ser. B 31 (1981), 111114. 2. B. Bollobas and A. Thomason, Random graphs of small order, in ``Random Graphs '83'' (M. Karonski and A. Rucinski, Eds.), Ann. Discrete Math., Vol. 28, pp. 4797, NorthHolland, Amsterdam, 1985.

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3. J. A. Bondy, Large dense neighbourhoods and Turan's theorem, J. Combin. Theory Ser. B 34 (1983), 109111. 4. J. A. Bondy, Erratum: Large dense neighbourhoods and Turan's theorem, J. Combin. Theory Ser. B 35 (1983), 80. 5. P. Erdo s, On the graph-theorem of Turan, Mat. Lapok 21 (1970), 249251. [In Hungarian] 6. P. Turan, An extremal problem in graph theory, Mat. Fiz. Lapok 48 (1941), 436452. [In Hungarian] 7. A. A. Zykov, On some properties of linear complexes, Mat. Sb. 24 (1949), 163188. [In Russian]