Star Saturation Number of Random Graphs

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1 Star Saturatio Number of Radom Graphs A. Mohammadia B. Tayfeh-Rezaie Schoo of Mathematics, Istitute for Research i Fudameta Scieces IPM, P.O. Box , Tehra, Ira ai m@ipm.ir tayfeh-r@ipm.ir Abstract For a give graph F, the F -saturatio umber of a graph G is the miimum umber of edges i a edge-maxima F -free subgraph of G. Recety, the F -saturatio umber of the Erdős Réyi radom graph G, p has bee determied asymptoticay for ay compete graph F. I this paper, we give a asymptotic formua for the F -saturatio umber of G, p whe F is a star graph. Keywords: Radom graph, Saturatio, Star graph. AMS Mathematics Subject Cassificatio 010: 05C35, 05C80. 1 Itroductio A graphs i this paper are assumed to be fiite, udirected, ad without oops or mutipe edges. The vertex set ad the edge set of a graph G are deoted by V G ad EG, respectivey. For ay subset S of V G, the iduced subgraph of G o S is deoted by G[S]. For a iteger 1 ad a rea umber p [0, 1], we deote by G, p the probabiity space of a graphs o a fixed vertex set of size where every two distict vertices are adjacet idepedety with probabiity p. I 1941, Turá posed oe of the foudatioa probems i extrema graph theory [8]. His questio was about the maximum umber of edges i a graph o vertices without a copy of a give graph F as a subgraph, a parameter which is ow deoted by ex, F. A dua idea caed saturatio umber was itroduced by Zykov [10] ad ater idepedety by Erdős, Haja, 1

2 ad Moo []. It asks for the miimum umber of edges i a edge-maxima F -free graph o vertices. We beow preset this otio i a more geera form. Fix a positive iteger ad a graph F. A graph G is caed F -saturated if G cotais o subgraph isomorphic to F but each graph obtaied from G by joiig a pair of o-adjacet vertices cotais at east oe copy of F as a subgraph. I other words, G is F -saturated if ad oy if it is a edge-maxima F -free graph. So, ex, F is equa to the maximum umber of edges i a F -saturated graph o vertices. The saturatio fuctio of F, deoted sat, F, is the miimum umber of edges i a F -saturated graph o vertices. For istace, it was proved by Erdős, Haja, ad Moo [] that sat, K r r r 1, where r ad K r is the compete graph o r vertices. For a give graph G, a spaig subgraph H of G is said to be a F - saturated subgraph of G if H cotais o subgraph isomorphic to F but each graph obtaied by addig a edge from EG \ EH to H has at east oe copy of F as a subgraph. The miimum umber of edges i a F -saturated subgraph of G is deoted by satg, F. Thus, sat, F is by defiitio equa to satk, F. We refer the reader to [3] ad the refereces therei for a survey o graph saturatio. I recet years, a ew tred i extrema graph theory has bee deveoped to exted the cassica resuts, such as Ramsey s ad Turá s theorems, to radom aaogues. The study reveas the behavior of extrema parameters for a typica graph. For istace, Korádi ad Sudakov iitiated the study of graph saturatio for radom graphs very recety [6]. They proved for every fixed p 0, 1 ad fixed iteger r 3 that sat G, p, K r 1 + o1 og 1 with high probabiity. Let us reca that, for a sequece X 1, X,... of radom variabes, we write X o1 with high probabiity if im P X ɛ 1, for ay ɛ > 0. Let K 1,r be the star graph o r +1 vertices. I this paper, we ivestigate the K 1,r -saturatio umber of G, p. The cassica versio was resoved

3 by Kászoyi ad Tuza [5], where they proved that sat, K 1,r r 1 r 8 r + r, if r + 1 3r, if 3r. The first o-trivia case, amey r, is especiay iterestig for the reaso that satg, K 1, is by defiitio equa to the miimum cardiaity of a maxima matchig i G. It has bee prove by Zito [9] that im P og 1 p < sat G, p, K 1, < og Here we show that with high probabiity sat G, p, K 1,r r 1 ; 1 + o1 r 1 og 1, for every fixed p 0, 1 ad fixed iteger r. Note that, for r, our resut gives a upper boud stroger tha 1 whereas our ower boud is weaker. It is fiay worth otig that for compete graphs the saturatio umber of radom graphs is much arger tha the cassica versio whie the parameter for star graphs is sighty smaer tha its cassica vaue. Resuts Let G be a graph ad k be a oegative iteger. A subset S of V G is caed k-idepedet if the maximum degree of G[S] is at most k. The k-idepedece umber of G, deoted by α k G, is defied as the maximum cardiaity of a k-idepedet set i G. I particuar, α 0 G αg is the usua idepedece umber of G. The foowig theorem is we kow ad is proved as Theorem 7.3 i [4]. Theorem.1. Matua [7] For ay fixed umber p 0, 1, with high probabiity. αg, p + o1 og 1 The foowig easy observatio ca be proved usig a straightforward uio boud argumet. We appy it to obtai a geeraized versio of Theorem.1. 3

4 Lemma.. Let X be a biomia radom variabe with parameters ad p 0, 1. The PX s s 1 p s for ay s {0, 1,..., }. Theorem.3. For every fixed umber p 0, 1 ad fixed iteger k 1, with high probabiity. α k G, p + o1 og 1 Proof. Let G G, p, q 1 p, ad b 1/q. For ay iteger s 1, et X s be the umber of iduced subgraphs i G o s vertices with at most sk/ edges. Ceary, X s 0 impies α k G s 1. For ay S V G with S s, et Y S cout the umber of edges i G[S]. By Lemma., EX s S V G S s s s P Y S ks ks q s ks e s e s s ks e se s k C s k q s s, ks q s ks k q s k 1 s for some fixed vaue C. Put s og b + k og b og b. We have og b s k q s og b + k og b s s ad so s k q s 0 as teds to ifiity. Therefore, EX s 0 ad sice PX s > 0 EX s by the Markov iequaity, it foows that PX s > 0 0 as goes to ifiity. This proves that α k G og b + k og b og b 1 with high probabiity. Now, the assertio foows from the fact α k G αg ad Theorem.1. The foowig emma is ater used to prove the ower boud o satg, K 1,r. Lemma.4. For every graph G o vertices ad iteger r, satg, K 1,r r 1 α r G. 4

5 Proof. Let H be a K 1,r -saturated subgraph of G. Let A be the set of vertices of H with degree at most r i H. Sice H is a K 1,r -saturated subgraph of G, every vertex i A V G\A is of degree r 1 i H ad G[A] H[A]. This impies that A α r G. We hece obtai that EH 1 deg H v r 1 α r G. v A We wi make use of the ext theorem i the proof of our mai resut. Theorem.5. Ao Füredi [1] Let G G, p be a radom graph ad H be a fixed graph o vertices with maximum degree, where +1 <. If 10 og p > +1 +1, the the probabiity that G does ot cotai a copy of H is smaer tha 1/. Now we are i the positio to prove our mai resut. Theorem.6. For every fixed umber p 0, 1 ad fixed iteger r, sat G, p, K 1,r r 1 with high probabiity. 1 + o1 r 1 og 1 Proof. Let G G, p, q 1 p, ad b 1/q. Usig Theorem.3 ad Lemma.4, we fid that im P satg, K 1,r r ɛr 1 og b 1, for ay ɛ > 0. So, it suffices to prove that im P satg, K 1,r r 1 1 ɛr 1 og b 1, for ay ɛ > 0. Fix ɛ ad et be the east iteger such that ɛ og b ad r 1 is eve. Aso, fix a reguar graph L o vertices with degree r 1. For ay S V G with S, et X S 1, if S is a idepedet set i G ad G[V G \ S] has a copy of L as a subgraph; 0, otherwise. 5

6 We assume to be arge eough wheever eeded. It foows from Theorem.5 that E[X S ] q. Therefore, if we et X S V G S X S, the E[X] q. Moreover, for every subsets S, T V G of size with S T i, we easiy see that E[X S X T ] q i. By the Chebyshev iequaity ad otig that goes to ifiity, we have PX 0 varx E[X] S,T V G S T i0 i0 E[X S X T ] E[X S ]E[X T ] E[X] S,T V G S T S T i i0 E[X S X T ] E[X S ]E[X T ] E[X] i i q i q q 1 q i i i q i i + i i q. i i i q i Usig the computatios give i the proof of Theorem 7.3 of [4], the ast summatio above coverges to 0 as ad hece PX 0 o1. This shows that with high probabiity there is S V G with S such i 6

7 that S is a idepedet set i G ad G[V G \ S] has a copy L of L as a subgraph. Deote the spaig subgraph of G with edge set EL by H. It is easiy see that H is a K 1,r -saturated subgraph of G ad EH r 1 r 1 1 ɛr 1 og b, which cocudes, as required. Ackowedgmets The authors woud ike to thak the aoymous referees for their hepfu commets ad correctios o a draft versio of this paper. Refereces [1] N. Ao ad Z. Füredi, Spaig subgraphs of radom graphs, Graphs Combi , [] P. Erdős, A. Haja, ad J.W. Moo, A probem i graph theory, Amer. Math. Mothy , [3] J.R. Faudree, R.J. Faudree, ad J.R. Schmitt, A survey of miimum saturated graphs, Eectro. J. Comb , #DS19. [4] A. Frieze ad M. Karoński, Itroductio to Radom Graphs, Cambridge Uiversity Press, Cambridge, 016. [5] L. Kászoyi ad Zs. Tuza, Saturated graphs with miima umber of edges, J. Graph Theory , [6] D. Korádi ad B. Sudakov, Saturatio i radom graphs, Radom Structures Agorithms , [7] D.W. Matua, The argest cique size i a radom graph, Techica Report, Departmet of Computer Sciece, Souther Methodist Uiversity, Daas, [8] P. Turá, Eie Extremaaufgabe aus der Graphetheorie, Hugaria Mat. Fiz. Lapok , [9] M. Zito, Sma maxima matchigs i radom graphs, Theoret. Comput. Sci , [10] A.A. Zykov, O some properties of iear compexes, Russia Mat. Sborik N.S ,