decompositions of

Short paths inε-regular pairs and small diameter decompositions of dense graphsJoanna Polcyn joaska@.plAndrzej Ruci´n ski rucinski@.plDepartment of Discrete MathematicsAdam Mickiewicz UniversityPozna´n,PolandAbstractIn this paper we give optimal optimal vertex degree conditions that guar-antee connection by short paths inε-regular bipartite graphs.We also studya related question of decomposing an arbitrary graph into subgraphs of smalldiameter.Keywords:ε-regular graphs,diameter,edge decomposition1IntroductionSince its discovery in late1970’s,the Szemer´e di Regularity Lemma has been a powerful tool in modern graph theory.Roughly speaking,this deep result from [16]provides a decomposition of every large graph into bipartite subgraphs most of which areε-regular pairs.Thus,naturally,there has been a growing interest in studying the properties of such graphs(see,e.g.,[1,2,3,4,5,6,7,10]).The goal of this paper is two-fold.First,in Section3we broaden our knowledge onε-regular pairs by determining the minimum degree of vertices that guarantees small diameter.It is quite easy to show(cf.Corollary3.3(i))that in anε-regular pair of order2n and density d≥εevery two vertices of degree at leastεn are con-nected by a short path.However,our main result in this direction(Theorem3.4) shows that this bound can be replaced by a quantity of order(ε2/d)n.Moreover, this is essentially optimal.The other result proved in this paper deals with graph decomposition into subgraphs of small diameter.Such decompositions have a potential application in distributed computing(see,e.g.,[12]).Here we approximate a given graph bya union of subgraphs with small diameter.It is easy to prove(cf.Proposition4.1),that for everyγ>0,the set of all but at mostγn2edges of every n-vertex graph G can be split into no more than1/γsubgraphs with diameter bounded from above by3/γ.In Theorem4.5,using the Szemer´e di Regularity Lemma,we1push the diameter down to four on the cost of an increase in the number of parts of the decomposition.Similar results for quasi-random3-uniform hypergraphs have been proved in[15]. 2PreliminariesLet G=(V,E)be a graph.For E0⊆E we denote by G[E0]=(V0,E0),where V0= e∈E0e,the subgraph of G induced by E0.Note that G[E0]does not need to be an(vertex)induced subgraph in the usual sense.By dist G(x,y)we denote the distance of vertices x,y∈V,that is,the length of a shortest path connecting them,if such a path exists.Otherwise we set dist G(x,y)=∞.By the diameter of G we mean diam(G)=max x,y∈V dist G(x,y).In particular, if G is not connected,then diam(G)=∞.For A⊆V,let N(A)be the set of all vertices adjacent in G to at least one vertex in A.In particular,if A={v}we write N(v).The number deg(v)=|N(v)| is the degree of vertex v in graph G.The minimum vertex degree is denoted by δG.In Section4we will use the inequality3|V|diam(G)<|U||W|.The number d G(U,W)is called the density of the graph G between U and W,or simply,the density of the pair(U,W).In the remainder of this section let G denote a bipartite graph with bipartition V=V1∪V2.A simple averaging argument yields the following fact.Fact2.1If d G(V1,V2)<d(resp.,d G(V1,V2)>d),then for all natural numbers ℓ1≤|V1|andℓ2≤|V2|there exist subsets U⊂V1,|U|=ℓ1and W⊂V2,|W|=ℓ2 with d G(U,W)<d(resp.,d G(U,W)>d).Note that eachε-regular graph isε′-regular for allε′≥ε.The last definition of this section deals with pairs of subsets of vertices with no edge in between.Definition2.3A pair of sets U⊆V1and W⊆V2with e G(U,W)=0is called a hole or a(|U|,|W|)-hole in G.A bipartite graph G is called(ℓ1,ℓ2)-holeless if there is neither an(ℓ1,ℓ2)-hole nor an(ℓ2,ℓ1)-hole in G.Note that,trivially,each(ℓ1,ℓ2)-holeless graph is also(ℓ′1,ℓ′2)-holeless for allℓ′i≥ℓi, i=1,2.Also,it follows from the definition ofε-regularity that,ifε≤d,then each (d,ε)-regular graph is(⌈ε|V1|⌉,⌈ε|V2|⌉)-holeless.However,it is proved in[13],that even much smaller holes are forbidden.Lemma2.4([13])(i)For all0<ε<d<1,d+ε≤1,there exists n0such that every(d,ε)-regularbipartite graph G with|V1|=|V2|=n≥n0is(ℓ,ℓ)-holeless for allℓ≥2ε(√d−εn+1.(ii)For all0<ε<d<1,d+ε<1,ε<(1−d)2/d and for allα<2ε(√d−εthere exists n0=n0(d,ε,α)such that for all n≥n0there exists a(d,ε)-regular graph with|V1|=|V2|=n containing an(⌈αn⌉,⌈αn⌉)-hole.3Short paths between vertices of sufficiently large degreesThis section is devoted to the problem of what minimum degree of vertices in anε-regular graph guarantees a small diameter.As a starting point recall an old result from[9]which says that almost all bipartite graphs with density d have diameter three.Although anε-regular pair resembles a random bipartite graph,in general nothing can be said about its diameter,because it may contain isolated vertices. On the other hand,it seems reasonable that vertices of sufficiently large degrees are connected by short paths.Indeed,below wefirst prove that degrees at least, roughly,(2ε3/2/√Proposition3.1Letℓ≥1and G be an(ℓ,ℓ)-holeless balanced bipartite graph.If deg(v)≥ℓand deg(w)≥ℓ,then dist G(v,w)≤4.Proof Without loss of generality we may assume that v∈V1.Considerfirst the case when w∈V2.By the assumption we have|N(v)|≥ℓand|N(w)|≥ℓ,and therefore,due to the absence of(ℓ,ℓ)-holes,there exists an edge e={x,y}with x∈N(v)and y∈N(w).Thus,vertices v,x,y,w form in G a path from v to w of length three(see Figure1).Considerfirst the case when v∈V1and w∈V2.Ifℓ≤⌈n/2⌉then there must be an edge between these two sets yielding a path of length at mostfive(see Figure 3).n+1d−εthen dist G(v,w)≤4.In particular,ifδG≥2ε(√εd−ε)/(d−ε)and sufficiently large n,there exists a(d,ε)-regular balanced bipartite graph G with2n vertices containing two vertices u,w∈V,deg(u)=deg(w)=⌊αn⌋,with dist G(u,w)≥5. Proof We only need to prove part(ii).By Lemma2.4(ii)there exists a(d,ε+ o(1))-regular graph with|V1|=|V2|=n−1containing an(⌈αn⌉,⌈αn⌉)-hole between sets A⊂V1and B⊂V2.We add two vertices,u and w,and all the edges connecting u with the vertices of B and w with the vertices of A.This way we obtain a new graph with the desired property.(We omit technical details here.)Again,the obtained bound is nearly optimal,namely there exist(d,ε)-regular graphs containing a vertex with degree only slightly smaller than2ε2n/(d+ε), which is disconnected from all vertices other than its neighbors.Theorem3.4(i)Let G be a(d,ε)-regular balanced bipartite graph with2n vertices,where0<ε≤1/2,ε<d<1,and n is sufficiently large.If2ε2nmin{deg(v),deg(w)}≥(|B|−|A|)(d+ε)|B|2<d+ε−αand note that0<γ≤ε−α.Let G′=(V′1∪V′2,E)be a(d+ε−γ,γ/2)-regular graph,where|V′1|=n−⌊αn⌋,|V′2|=n−1and n is sufficiently large.The existence6of such graphs can be proved easily using random graphs (see [13]).Notice that 0<d +ε−γ<1.Then,we add to V ′1a set A of ⌊αn ⌋vertices and a vertex w to V ′2,adjacent to all vertices of A .We claim that the graph G =(V 1∪V 2,E ),where V 1=V ′1∪A and V 2=V ′2∪{w },has the desired property.Since G [A ∪{w }]is an isolated star in G ,it remains to show that G is (d,ε)-regular.Let U ⊂V 1,W ⊂V 2,|U |,|W |≥εn .Set U ′=U \A and W ′=W \{w }and observe that |U ′|≥(ε−α)n >(γ/2)|V ′1|,and,for large n ,|W ′|>(γ/2)|V ′2|(see Figure4).2γ+O (n −1)<d +ε,and d G (U,W )≥d G ′(U ′,W ′) 1−|A ||W | > d +ε−3ε−O (n −1)>(d +ε−2γ) 1−2εε =d −ε+γ(d +ε)d +ε+2γd +ε+2γε.4An approximate decomposition into few sub-graphs with small diameterThis section provides two results about approximating a given graph by a union of subgraphs with small diameter.As a consequence of Proposition4.1below,for a givenγ>0,we can split the set of all but at mostγ|V(G)|2edges of G into no more than1/γsubgraphs with diameter bounded from above by3/γ.Then in Theorem4.5,using the celebrated Szemer´e di Regularity Lemma,we decrease the bound on the diameter to four,at the price of increasing the number of subgraphs in the partition.Thus,in a sense,every graph can be decomposed into a bounded number of subgraphs with small diameter,provided a small set of edges and/or vertices can be ignored.Proposition4.1Let0<γ<1.The set of edges of every graph G=(V,E)with |V|=n can be partitioned into k+1subsets,E=E0∪E1∪···∪E k,where k≤1/γand|E0|≤γn2,in such a way that for each1≤s≤k we have diam(G[E s])≤3/γ.Proof Keep removing from G,one by one,vertices of degree less thanγn,until the minimum degree of the obtained induced subgraph G′is at leastγn.Note that we have removed at mostγn2edges.Since for each connected component C of G′we have|V(C)|>δC≥γn,there are,obviously,at most1/γcomponents. Moreover,for each such component C,by(1)we havediam(C)<3|V(C)|γn=3Note that,in fact,the above proof yields a partition into vertex-disjoint sub-graphs.In Proposition4.1both bounds,on the diameter and on the number of sub-graphs G[E s],depend linearly on1/γ,and thus grow to infinity when the precision of approximation,γ,tends to zero.However,one can compromise on one of these bounds,improving the other to the extent that it becomes independent ofγ.In-deed,using the Szemer´e di Regularity Lemma[16]and Corollary3.3(ii),we will put the cap of four on the diameter,at the cost of letting the number of subgraphs in the partition to be an enormous constant.Before we make this precise,let us quote the Regularity Lemma.Definition4.2Let0<ε<1,t be a positive integer,and let G=(V,E)be a graph.We call a partition V=V0∪V1∪···∪V tε-regular,if|V1|=|V2|=···=|V t|, |V0|<t,and all but at mostε t2 pairs(V i,V j),1≤i<j≤t,areε-regular. Theorem4.3(Szemer´e di Regularity Lemma,[16])Let0<ε<1and let t0be a positive integer.There exist integers N=N(ε,t0)and T=T(ε,t0)such that every graph G=(V,E)with|V|≥N vertices admits anε-regular partition V=V0∪V1∪···∪V t with t0≤t≤T.8In the proof of Theorem4.5below,we will also need a simple observation that forε≤d/3,every(d,ε)-regular graph can be approximated by a subgraph of diameter at most four.Fact4.4If0<3ε≤d<1then G contains an induced subgraph G′such that |E(G)\E(G′)|≤2εn2and diam(G′)≤4.Proof For each i=1,2,letU i={v∈V i:deg(v)≤2εn}.Since3ε≤d,it follows easily from the(d,ε)-regularity of G that|U i|<εn,i=1,2. Let W i be such that U i⊆W i⊆V i and|W i|=⌊εn⌋:=m,i=1,2.Finally,let G′=G−(W1∪W2)be the graph obtained from G by removing all vertices of W1∪W2,together with their incident edges.Let E0be the set of all removed edges.Note that G′=G[E\E0]and that |E0|≤(2m)n≤2εn2.Moreover,it is easy to see that G′is(d,ε′)-regular,where ε′=εn/(n−m)<d,whileδG′≥2εn−εn=εn=ε′(n−m)≥...n.Therefore,by Corollary3.3(i),diam(G′)≤4.n2.5Let F2be the set of all edges of G contained in one of the sets V i,i=1,2,...,t. We have⌊n/t⌋2 <n22t0≤γ|F2|≤tt 2<ε10n2.9Finally,let F 4be the set of all edges belonging to the pairs (V i ,V j )with density d G (V i ,V j )≤4ε.Then|F 4|≤ t 2 4ε n 5n 2.Consequently,setting E ′0=F 1∪F 2∪F 3∪F 4,we have|E ′0|<γ10n 2+γ5n 2=4t 2=2t 2,such that diam(G [E ′s \E 0s ])≤4.Altogether we have deleted a set E ′′0=ks =1E 0s of at most t 2 2t2<15γn 2+1Note that the proofs of Proposition 4.1and Theorem 4.5both yield algorithms of complexity O (n 2)which construct the respective partitions.The latter is based on a O (n 2)-time algorithm found in [11]which builds an ε-regular partition in every n -vertex graph.Acknowledgements:We would like to thank both Referees for their valuable comments on the presentation of our paper.Most importantly,Referee A suggested to look at the problem of minimum degree guaranteeing paths of length at most four (Corollary 3.3)and sketched a solution to it.He also supplied us with an alternative construction for the proof of Theorem 3.4(ii)and observed the algorithmic aspects of Proposition 4.1and Theorem 4.5discussed in the previous paragraph.References[1]N.Alon,R.A.Duke,H.Lefmann,V.R¨o dl &R.Yuster,The algorithmicaspects of the regularity lemma,Journal of Algorithms 16(1)(1994)80-109.[2]N.Alon,V.R¨o dl &A.Ruci´n ski,Perfect matchings in ε-regular graphs TheElectr.J.of Combin.5(1)(1998)#R13.10[3]A.Czygrinow&B.Nagle,Matrix-free proof of a regularity characterization,bin.10(2003)#R39.[4]A.Czygrinow&B.Nagle,Strong edge colorings in uniform graphs DiscreteMathematics,286(3)(2004)219-223.[5]R.A.Duke,H.Lefmann&V.R¨o dl,A fast approximation algorithm for com-puting the frequencies of subgraphs in a given graph.SIAM put.24 (1995)598-620.[6]A.Frieze,On the number of perfect matchings and Hamilton cycles in epsilon-regular non-bipartite graphs Electr.J.of Combin.7(2000)R57.[7]A.Frieze&M.Krivelevich,On packing Hamilton cycles in epsilon-regulargraphs J.of Combin.Th.B94(2005)159-172.[8]S.Janson,T. 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