Minimum degree conditions for large subgraphs

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<ul><li><p>Minimum degree conditions for large subgraphs</p><p>Peter Allen 1</p><p>DIMAPUniversity of Warwick</p><p>Coventry, United Kingdom</p><p>Julia Bottcher and Jan Hladky 2,3</p><p>Zentrum MathematikTechnische Universitat MunchenGarching bei Munchen, Germany</p><p>Oliver Cooley 4</p><p>School of MathematicsUniversity of Birmingham</p><p>Birmingham, United Kingdom</p><p>Abstract</p><p>Much of extremal graph theory has concentrated either on nding very small sub-graphs of a large graph (such as Turans theorem [10]) or on nding spanning sub-graphs (such as Diracs theorem [3] or more recently work of Komlos, Sarkozy andSzemeredi [7,8] towards a proof of the Posa-Seymour conjecture). Only a few resultsgive conditions to obtain some intermediate-sized subgraph. We contend that thisneglect is unjustied. To support our contention we focus on the illustrative case ofminimum degree conditions which guarantee squared-cycles of various lengths, butalso oer results, conjectures and comments on other powers of paths and cycles,generalisations thereof, and hypergraph variants.</p><p>Keywords: extremal graph theory, minimum degree, large subgraphs</p><p>Electronic Notes in Discrete Mathematics 34 (2009) 7579</p><p>1571-0653/$ see front matter 2009 Elsevier B.V. All rights reserved.</p><p></p><p>doi:10.1016/j.endm.2009.07.013</p></li><li><p>1 Introduction</p><p>We recall that the classic theorems of Turan [10] and Erdos and Stone [4]provide best possible (in the latter case, up to an o(n) error term) minimumdegree conditions (and indeed density conditions) for a large graph G to con-tain respectively a clique and a xed small subgraph H. In the rst theoremthe extremal graphs are Turan graphs; in the second they are Turan graphswith at most o(n2) edges changed.</p><p>We dene the bottle graph Bk(n, ) to be a complete k-partite graph on nvertices whose smallest part is as small as possible subject to the minimumdegree of the graph being . We note that when = k1</p><p>kn the smallest part</p><p>is at most one vertex smaller than the largest, and we obtain the Turan graphTk(n).</p><p>Diracs theorem [3] provides a best possible minimum degree condition fora graph to contain a Hamilton cycle; the extremal graphs are bottle graphs.</p><p>We dene the k-th power of a graph F , F k, to be that graph with vertexset V (F k) = V (F ) and edge set E(F k) = {uv: distF (u, v) k}. GeneralisingDiracs theorem, Posa (for k = 2) and Seymour (for general k) gave conjec-tured best possible minimum degree conditions for a graph G to contain thek-th power of a Hamilton cycle; these conjectures have recently been provedfor large graphs by Komlos, Sarkozy and Szemeredi [7,8]. Again the extremalgraphs are bottle graphs.</p><p>We say a graph G contains an H-packing on v vertices when there exist in Gvertex-disjoint copies of the graph H covering in total v vertices of G. Hajnaland Szemeredi [5] gave for each k best possible minimum degree conditionsfor a graph G to contain a perfect Kk-packing (i.e. a packing covering all butat most k 1 vertices of G). The extremal graphs are bottle graphs.</p><p>Finally, Kuhn and Osthus [9] gave for each xed H best possible (up toan o(n) error term) minimum degree conditions for an n-vertex graph G tocontain a perfect H-packing. Yet again, their extremal graphs are obtained</p><p>1 Email: . Supported by DIMAPCentre for Discrete Math-ematics and its Applications at the University of Warwick.2 Email: . Partially supported by DFG grant TA 309/2-1.3 Email: . Supported in part by the grants GAUK 202-10/258009 andDAAD.4 Email: .</p><p>P. Allen et al. / Electronic Notes in Discrete Mathematics 34 (2009) 757976</p></li><li><p>by changing at most o(n2) edges of suitable bottle graphs.</p><p>2 Large subgraphs</p><p>If one desires to nd in an n-vertex graph G not a perfect H-packing but onlya partial H-packing, then for H = Kk the Hajnal-Szemeredi theorem alreadygives a best possible minimum degree condition:</p><p>Theorem 2.1 (Hajnal, Szemeredi) If G is any n-vertex graph with minimumdegree (G) (k2</p><p>k1n,k1k</p><p>n) we are guaranteed a Kk-packing covering at leastthe number of vertices covered by a maximal Kk-packing in Bk(n, ).</p><p>In 2000 Komlos [6] generalised this (albeit with an o(n) error term) to gen-eral partial H-packings; as with the result of Kuhn and Osthus the statementis not short, but amounts to a linear relationship (up to o(n) error) betweenthe number of vertices guaranteed in an H-packing and the minimum degree;the only real surprise with H-packings is that for some H there is a plateau;that is, an (n)-sized interval of values of over which we are guaranteed thatif (G) = then G has an almost-perfect packing but we are not guaranteeda perfect packing. The extremal graphs for H-packings are not necessarilybottle graphs, but they dier only in at most o(n2) edges.</p><p>We wish to give a similar generalisation of Posas conjecture. We note thatit is straighforward to extend Komlos, Sarkozy and Szemeredis proof to:</p><p>If G is a suciently large n-vertex graph which has minimum degree(G) 2n/3 then G contains C23 for each 3 [3, n], and if the inequal-ity is strict also C for each {3, 4} [6, n]. It is this stronger statementthat we will generalise (for large graphs).</p><p>We wish to nd a (best possible) function sc(n, ) such that any n-vertexgraph G with minimum degree (G) contains C23 for each 3 sc(n, ).There are two reasonable conjectures one might makeboth false.</p><p>The rst is that the extremal graphs should (yet again) be bottle graphsand thus we should nd sc(n, ) 6 3n. This is far too optimistic.</p><p>The second is that the method used for previous spanning subgraph andpartial packing results should work: start a construction at any vertex andcontinue cleverly until stuck, when the best possible result will be achieved.This would suggest that sc(n, ) 3 3n/2 (provided = n/2 + (n))which is almost always far too pessimistic.</p><p>In fact, the extremal graphswhich we call SC(n, )are far from beingbottle graphs, and the function sc(n, ) is far from linear: it has (as n grows) anunbounded number of jumps; that is, places where when increases by one,</p><p>P. Allen et al. / Electronic Notes in Discrete Mathematics 34 (2009) 7579 77</p></li><li><p>sc(n, ) increases by (n). An interesting feature of the extremal graphs isthateven though we cannot guarantee C2 G for any 4 not divisible by3, since G could be tripartitethey do in fact contain C2 for every sc(n, ).One might think that excluding all, or at least one, of these extra squared-cycles with chromatic number four would not change the result by very muchbut this is again false. We will describe the graphs SC(n, ) and show that,when = n/2+(n), sc(n, ) is equal to the length of the longest squared-cycleof SC(n, ), proving the following theorem.</p><p>Theorem 2.2 (Allen, Bottcher, Hladky) Given any &gt; 0 there is n0 suchthat the following hold for any graph G on n n0 vertices with(G) = (n/2 + n, 2n/3).</p><p>First, C23 G for each 3 sc(n, ). Second, either C2 G for each {3, 4} [6, sc(n, )], or C23 G for each 3 6 3n n.</p><p>We conjecture that the n error term can be removed from the secondcondition, but it is necessary that (G) is not too close to n/2. Our resultsgive also best-possible conditions for squared-paths.</p><p>Surprisingly, even for simple cycles no such best possible theorem has pre-viously been given. We give the correct result.</p><p>Theorem 2.3 (Allen) Given an integer k 2 there is n0 = O(k20) such thatthe following hold for any graph G on n n0 vertices with (G) = n/k.</p><p>First, C G for every even 4 nk1. Second, either C G foreach 2n</p><p> 1 n</p><p>k1, or C G for every even 4 2.</p><p>3 Further work</p><p>At the time of writing the correct generalisation of Theorem 2.2 to higherpowers of paths or cycles remains a conjecture. However there are furtherquestions in this areaBollobas and Komlos suggested a further generalisationof the Seymour conjecture to objects more complicated than powers of cycles(which was recently proved by Bottcher, Schacht and Taraz [1]); we believethat the related large subgraph questions exhibit interesting phenomena whichare not found with powers of cycles.</p><p>Finally, there has recently been progress in understanding extremal phe-nomena in r-uniform hypergraphs; unsurprisingly the situation here is muchmore complex than with simple graphs (there are for example multiple rea-sonable generalisations of the notion of minimum degree). Nevertheless, weoer some observations on matchings and paths in this setting.</p><p>P. Allen et al. / Electronic Notes in Discrete Mathematics 34 (2009) 757978</p></li><li><p>References</p><p>[1] Bottcher, J., M. Schacht and A. Taraz, Proof of the bandwidth conjecture ofBollobas and Komlos, Math. Ann 343(1) (2009), 175205.</p><p>[2] Corradi, K. and A. Hajnal, On the maximal number of independent circuits ofa graph, Acta Math. Acad. Sci. Hungar. 14 (1963), 423443.</p><p>[3] Dirac, G.A., Some theorems on abstract graphs, Proc. London Math. Soc. s3-2(1952), 6981.</p><p>[4] Erdos, P. and A. Stone, On the structure of linear graphs, Bull. Amer. Math.Soc. 52 (1946), 10871091.</p><p>[5] Hajnal, A. and E. Szemeredi, Proof of a conjecture of Erdos, Combinatorialtheory and its applications, Vol. 2, North-Holland (1970), 601623.</p><p>[6] Komlos, J., Tiling Turan theorems, Combinatorica 20 (2000), 203218.</p><p>[7] Komlos, J., G. N. Sarkozy and E. Szemeredi, On the square of a Hamiltoniancycle in dense graphs,, Random Struct. Algorithms 9 (1996), 193-211.</p><p>[8] Komlos, J., G. N. Sarkozy and E. Szemeredi, Proof of the Seymour Conjecturefor large graphs, Ann. Comb. 2 (1998), 4360.</p><p>[9] Kuhn, D. and D. Osthus, The minimum degree threshold for perfect graphpackings, Combinatorica, to appear.</p><p>[10] Turan, P., On an extremal problem in graph theory (in Hungarian), MatematikoFizicki Lapok 48 (1941), 436452.</p><p>P. Allen et al. / Electronic Notes in Discrete Mathematics 34 (2009) 7579 79</p></li></ul>


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