Minimum congestion spanning trees in bipartite and random graphs

Transcription

1 Minimum congetion panning tree in bipartite and random graph M.I. Otrovkii Department of Mathematic and Computer Science St. John Univerity 8000 Utopia Parkway Queen, NY 11439, USA September 14, 007 Abtract. The firt problem conidered in thi paper: i it poible to find upper etimate for the panning tree congetion for bipartite graph which are better than for general graph? It i proved that there exit a bipartite verion of the known graph with panning tree congetion of order n 3, where n i the number of vertice. The econd problem i to etimate panning tree congetion of random graph. It i proved that the tandard model of random graph cannot be ued to find graph whoe panning tree congetion ha order greater than n 3. Keyword: Bipartite graph; random graph; minimum congetion panning tree. 000 Mathematic Subject Claification: 05C05 Abbreviated title: Minimum congetion panning tree 1

2 1 Introduction In thi paper we conider finite imple graph. Our graph-theoretic terminology follow [1]. For a graph G by V G and E G we denote it vertex et and it edge et, repectively. Let G be a graph and let T be a panning tree in G that i, T i both a tree and a panning ubgraph of G). For each edge e of T let A e and B e be the vertex et of the component of T \e. By e G A e, B e ) we denote the number of edge with one end vertex in A e and the other end vertex in B e. We define the edge congetion of G in T by ecg : T ) = max e E T e G A e, B e ). The name come from the following analogy. Imagine that edge of G are road, and that edge of T are thoe road which are cleaned from now after nowtorm. For each road g E G there exit a unique path P g in T joining the end vertice of g, we call uch path a detour, even in the cae when P g = g. For an edge h of T it i quite natural to define the congetion ch) a the number of time h i ued in different detour {P g } g EG. Then ecg : T ) = max h E T ch). It i clear that for application it i intereting to find a panning tree which minimize the congetion. We define the panning tree congetion of G by G) = min{ecg : T ) : T i a panning tree of G}. 1) Each panning tree T in G atifying ecg : T ) = G) i called a minimum congetion panning tree for G. The parameter ecg : T ) and G) were introduced and tudied in []. Thee parameter are of interet in the tudy of Banach-pace-theoretical propertie of Sobolev pace on graph, ee [3, Section 3.5.1]. An additional motivation for thi tudy i that minimum congetion panning tree can be conidered a congetional analogue of the well-known hortet or minimal) panning tree. See [4] for an account on hortet panning tree, and on other minimality propertie of panning tree. One of the natural quetion about G) i to find the maximal poible value µn) of G) for graph G with n vertice. The rate of growth of the function µn) wa tudied in Section 3 of [], where i wa hown that c 1 n 3 µn) c n for ome abolute contant c 1 and c, and ome etimate for c 1 and c were found. The firt problem conidered in thi paper i: i it poible to find upper etimate of G) for bipartite graph which are better than for general graph? In connection with thi problem we prove that there exit a bipartite verion of the graph with panning tree congetion of order n 3 from [, Theorem ], but it ha to have a larger radiu than the example from []. In Section 3 we tudy panning tree congetion of random graph. A i well known random graph can be ued to find example with cloe-to-extremal behavior for many problem. We prove that the tandard model of random graph cannot be ued to improve the exponent 3 in the lower etimate for µn).

3 Minimum congetion panning tree in bipartite graph In [, Theorem ] it wa hown that there exit graph G with n vertice and G) cn 3, where c > 0 i an abolute contant. Our firt purpoe i to how that with a bit wore etimate for c) the ame reult remain true for bipartite graph. We refer to [5] for a ytematic expoition of the theory of bipartite graph. We ue the following way of getting a bipartite graph from an arbitrary graph, we call it duplication. Let H be a graph with vertex et V H. By duplication of the graph H we mean the bipartite graph H defined in the following way. We replace each vertex v V H by a pair of vertice u v, w v, and let V H = {u v, w v : v V H }. The vertex u v i adjacent with w x in H if an only if x V H i either v or a vertex adjacent to v. It i clear that H i a bipartite graph and that the et {u v } v VH ; {w v } v VH form a bipartition. We are going to how that the panning tree congetion of the duplication of the graph G conidered in [, Theorem ] atifie the ame etimate below a G). Recall the definition of G: Let n be of the form n = 3k k, where k i uch that k i an integer. We alo aume that k > 4. The graph G ha n vertice and i defined in the following way. We repreent V G a a union of 3 et, V G = C 1 C C 3, where C 1 = C = C 3 = k, C 1 C = C C 3 = k, and C 1 C 3 =. The edge et of G i defined in the following way: vertice v and u in G are adjacent if and only if u, v C i for ome i {1,, 3}. Theorem 1 Denote by G the duplication of the graph G. Then G i a bipartite graph with n = 6k 4 k vertice and G) 1 k3/. Remark. Our proof follow the line of the proof of Theorem in [], with modification needed becaue of the duplication. Proof. Working with tree it will be convenient to ue related definition and obervation that are going back to C. Jordan [6], ee [7, pp ]. Let u be a vertex of a tree T. If we delete all edge incident with u from T, we get a foret. The maximal number of vertice in component of the foret i called the weight of T at u. A vertex v of T i called a centroid vertex if the weight of T at v i minimal. Each tree ha one or two centroid vertice. Let T be a panning tree in G atifying G) = ec G : T ). Let c be a centroid of T, if T ha two centroid we chooe one of them, and fix the notation c for it. Denote the et {w v } v Ci and {u v } v Ci by D i and E i, repectively, let C i = D i E i. Since the et C 1 and C 3 are dijoint, the centroid c doe not belong to at leat one of them. Without lo of generality we aume c / C 1. The edge incident with c in T will be called central edge. If we remove all central edge from T, we get a foret. Let V 1,..., V t be vertex et of connected component of the foret, except the component whoe only vertex i c. 3

4 There i a natural bijective correpondence between the et of central edge and the et {V 1,..., V t }. Let u denote the central edge correponding to V i by e i. Oberve that C 1 cannot interect more than k of the et V 1,..., V t. In fact, only k vertice of C1 are adjacent to vertice that are not in C 1, o there are only k entrance into C 1. Since C 1 contain k vertice, thi implie that there exit j {1,..., t} uch that the interection V j C 1 ha at leat k vertice. Let γ 1 = V j C 1. Then γ 1 = δ 1 + ɛ 1, where δ 1 = V j D 1 and ɛ 1 = V j E 1. There are δ 1 k ɛ 1 ) + ɛ 1 k δ 1 ) = γ 1 k ɛ 1 δ 1 edge joining V j C 1 with C 1 \V j. Alo ɛ 1 δ 1 γ 1 4. Hence e j i ued in at leat k γ ) 1 γ 1 detour. Since γ 1 k and k > 4, then either k γ ) 1 γ 1 1 k3/ or k γ 1 < 1 k. In the former cae we are done. In the latter cae γ 1 = ɛ 1 + δ 1 > k k, hence at mot k of the k element of the et C 1 C are not in V j, and we get C 1 C V j > k. Therefore C V j > k. Let γ = C V j. Arguing in the ame way a above, there are more than k γ ) γ edge joining V j C with C \V j. Hence e j i ued in at leat k γ ) γ detour. Since γ > k and k > 4, then either k γ ) γ 1 k3/ or k γ < 1 k. In the former cae we are done. In the latter cae we get V j > C 1 + C C 1 C C 1 \V j C \V j = 4k 4 k. If k > 4, thi inequality implie V j > V G /. We get a contradiction with the fact that c i a centroid. It would be intereting to find out for general graph: how doe the duplication affect the parameter G)? In particular, we ugget the following problem. Problem 1 Let G be a connected graph and G be the graph obtained by duplication of G. I it true that G) G)? Or, at leat, G) c G) for ome abolute contant c > 0?) Recall ee [1, p. 10]) that the radiu of a graph G i min v max d Gv, w), where d G i the w tandard metric on G. Oberve that the graph from [, Theorem ] decribed above ha radiu, and it duplication ha radiu 3. The following reult how that for bipartite graph of radiu the quantity G) cannot be large. Theorem Let G be a bipartite graph of radiu. Then G) 3 V G. Proof. Conider a vertex v in G atifying d G u, v) for every u V G. 4

5 Let B = {u V G : d G u, v) = 1}. Let m = B and B = {b 1,..., b m }. Let A = {u V G : d G u, v) = }. We conider a partition {A i } m i=1 of A atifying the condition All vertice from A i are adjacent to b i. The um m i,j=1 A i A j ha the minimal poible value. Let T be the tree with the following edge et E T = {vb i : i = 1,..., m} {b i a : a A i, i = 1,..., m}. By uv we denote the edge joining vertice u and v.) It i eay to ee that T i a panning tree. Let u etimate ecg : T ). It i eay to ee that an edge of the form b i a i ued in d a n 1) detour d a i the degree of a). So it remain to etimate the number of detour uing an edge of the form vb i. If U and W are et of vertice of G, we ue the notation E G U, W ) for the et of edge with one end vertex in U and the other end vertex in W. The edge vb i i ued in detour for )) D := E G {b i } A i, {v} {b j } j i A j j i = {b i v} E G {b i }, j i A j ) E G A i, {b j } j i ). Since G i bipartite, there are no edge between different vertice from A and no edge between different vertice from B.) Let n = V G. Then D 1 + n m + 1)) + E G A i, {b j } j i ). Obervation. A vertex b t i not adjacent to any vertice from A if A atifie A > A t + 1. In fact, otherwie we chooe a vertex from A which i adjacent to b t, and move it to A t. We get a new partition {A k } atifying A = A 1, A t = A t +1, and A k = A k for k / {, t}. It i eay to ee that m A i A j m < A i A j. i,j=1 i,j=1 By the obervation n m + 1) E G A i, {b j } j i ) A i {j : A j A i 1} A i A i 1 n m + 1)), if A i > 1. We get an etimate E G A i, {b j } j i ) m 1 if A i = 1. Hence D max{3n 3m, n 1}. Therefore ecg : T ) < 3n = 3 V G. 5

6 3 Minimum congetion panning tree in random graph Let Gn, p) be the tandard probability pace of graph ee [1, p. 17]). We denote the tandard probability on thi pace by P p. Our purpoe i to how that the panning tree congetion of graph in Gn, p) i, with high probability, On 3 ). Our approache to howing that the panning tree congetion i mall are different in the cae p n 1 and p n 1. In the firt cae the random graph jut doe not have enough edge to have large congetion. In the econd cae the graph ha, with high probability, nice panning tree ee Theorem 3 for precie meaning of thi tatement), whoe exitence implie the deired bound for the congetion. Firt we conider the cae of mall p. Let A = A n,p be the interection of the following two event in Gn, p): {G n,p i connected} and {G n,p : G n,p ) n 3 }. Propoition 1 If p n 1, then P p A) exp 3 ) 16 n 3. Proof. We ue the following traightforward etimate G) E G V G + ee [, Theorem 1a)]). Hence P p A) P p {G n,p ha at leat n 3 edge}. The expected number λ of edge of G n,p i nn 1) p. If p n 1, then λ < n 3. By a Chernoff-type bound ee [8, Formula.5), p. 6]), P p {G n,p ha at leat n 3 edge} exp n 3 λ) ) λ + n 3 λ 3 exp n 3 /) ) = exp 3 ) 16 n 3. 3 n 3 Now let n 1 p 1 and let n N be fixed the dicuion below) aume, alo,. We aume n 1 that n i not very mall). Let = 1pn 1), and d = log that n i large enough o that. A tree T will be called a nice tree of depth k, k {0, 1,,..., d} correponding to the pair p, n) if it ha a vertex v 0 of degree, at mot n vertice, and the cardinalitie of the vertex et of the component of T obtained after removal of v 0, we call uch component branche, atify the following condition: If 0 k n 1 log ), then each of the branche ha k vertice. ) n 1 n 1 If log i not an integer, and k = log ha between k 1 and k vertice, and the tree ha n vertice. ), then each of the branche 6

7 Denote by T k Gn, p) the event of exitence of a ubgraph which i a nice tree of depth k. Theorem 3 For each k {0,..., d} the etimate P p T k ) 1 exp c 1 n)) k+1 1 exp c n) hold, where c1 > 0 and c > 0 are abolute contant. Remark. To relate thi theorem with our tudy of minimum congetion panning tree we oberve that a nice tree of depth d ha the following propertie: It i a panning tree. If we delete any of it edge, one of the obtained component ha at mot d n 1 vertice. Hence the edge i ued in at mot n n 1 detour. Therefore thee condition imply for each graph in which T i a panning tree) that ecg : T ) n n 1 Cn 3. Proof of Theorem 3. We etimate P p T k ) uing induction. Oberve that T 0 i the event of exitence of a vertex whoe degree i at leat. The high probability of thi event follow from a Chernoff-type etimate. In fact, the expected degree of a vertex in pn 1). Hence for a fixed vertex v 0 whoe degree we denote by d v0 ) the probability α := P p {d v0 1 pn 1)} can be etimated uing [8, inequality.6), p. 6]. We get α exp 1 ) pn 1)) = exp 18 ) pn 1) pn 1) exp n 1 ) 8 n and P p T 0 ) 1 exp n 1 ) 8 1 exp c 1 n). n To etimate P p T k ) for k > 0 we ue conditional probabilitie: P p T k ) = P p T k 1 ) P p T k T k 1 ). We etimate P p T k T k 1 ) uing the following lemma. Recall that a et of edge i called a matching if no two of them have a common vertex. Lemma 1 Let A and B be dijoint vertex et atifying A n and B n. Then the probability that G n,p doe not contain a matching F A, uch that each edge from F A ha one end vertex in A and one end vertex in B, and each vertex from A i incident with exactly one edge from F A, i exp c 1 n) for ome abolute contant c1 > 0. Proof. It i clear that the probability i the leat if A = B = n. In uch a cae the probability can be etimated uing the method from B. Bollobá and A. Thomaon [9, pp ] ee, alo, [10, pp ] and [8, pp. 8 84]). Since in our cae p n 1, we can ue more crude and imple etimate. We denote n by m and write the etimate from [9] with m intead on n. We need a verion of the following lemma from [9, Lemma 6, p. 64]. A i mentioned in [9], it can be eaily derived from Hall theorem which can be found, e.g, in [1, p. 77]). 7

8 Lemma [9]) Let G be a bipartite graph with vertex clae V 1 and V, V 1 = V = m. Suppoe that G doe not have a matching F, uch that each of the vertice i incident with exactly one edge from F. Then there i a ubet N V i i = 1, ) uch that i) ΓN) := {y : y i adjacent to ome x N} ha N 1 element. ii) 1 N m + 1)/. We conider the bipartite ubgraph of G n,p whoe vertex clae are V 1 = A and V = B the edge et of thi ubgraph conit of all edge of G n,p having one end vertex in A and one end vertex in B). Denote by I a a = 1,,..., m + 1)/ ) the event that there i a et N V i i = 1, ), N = a atifying i) and ii) for thi ubgraph. Since we have m a ) choice for N with N = a and m a 1) more choice for ΓN), we get ) ) m m P p I a ) 1 p) am+1 a) a a 1 Uing the elementary etimate ) m a m a and 1 p < e p, we get P p I a ) m a exp pam + 1 a)) m exp p m + 1 m + 1 Let m 1 =. By Lemma the probability of non-exitence of a et F A atifying m1 ) the condition of Lemma 1 can be etimated from above by P p I a We have the etimate m1 P p a=1 ) m 1 I a a=1 a=1 )) a m exp p m + 1 )) a m exp p m+1) 1 m exp p m+1) exp c 1n 1 ) for ome abolute contant c 1 and ufficiently large n. Of coure, we can elect the ame contant here and in our etimate for P p T 0 ).) We continue our proof of the theorem. Suppoe that we have proved the etimate for T k 1. If a nice tree of depth k 1) ha n vertice, there i nothing more to prove. Otherwie it ha k vertice. Oberve that for each et of k vertice in G n,p the exitence of a nice tree with thi vertex et i independent from the exitence of edge between vertice from thi et and the remaining vertice. Let T be a nice tree of depth k 1). We conider two cae. Cae 1: k + 1 n. In thi cae there are at leat k 1 vertice in G n,p which are not vertice of T. Let A be the union of the vertex et of all branche of T and B be the complement of the vertex et of T. If we add to T a matching F A atifying the condition of Lemma 1 and it vertice), we get a nice tree of depth k, hence P p T k T k 1 ) P p exitence of F A T k 1 ) = P p exitence of F A ) 1 exp c 1 n 1 ) 8

9 we ued the independence mentioned above). Hence P p T k ) P p T k 1 )1 exp c 1 n 1 )) in thi cae. Cae. k + 1 > n. In thi cae k = d, and the number of vertice in G n,p which are not in T i < k 1. We denote thi et by A and let B the et of all vertice of T except v 0. We apply Lemma 1 to thee A and B, and denote the correponding matching by F A. Adding F A and it vertex et) to T we get a nice tree of depth d. The probability i etimated in the ame way a in Cae 1. Problem It would be intereting to find more precie etimate for the panning tree congetion of random graph. It eem plauible that mot random graph in Gn, p) atify G n,p ) C n for ome abolute contant C. Reference [1] B. Bollobá, Modern Graph Theory, New York, Springer-Verlag, [] M. I. Otrovkii, Minimal congetion tree, Dicrete Math., ), [3] M. I. Otrovkii, Sobolev pace on graph, Quaetione Mathematicae, 8 005), [4] B. Y. Wu and K.-M. Chao, Spanning tree and optimization problem, Boca Raton, Chapman & Hall/CRC, 004. [5] A. S. Aratian, T. M. J. Denley, and P. Häggkvit, Bipartite graph and their application, Cambridge Tract in Mathematic, 131, Cambridge, Cambridge Univerity Pre, [6] C. Jordan, Sur le aemblage de ligne, J. Reine Angew. Math., ), [7] F. Harary, Graph Theory, Addion-Weley Publihing Company, [8] S. Janon, T. Luczak, and A. Rucińki, Random graph, New York, Wiley, 000. [9] B. Bollobá and A. Thomaon, Random graph of mall order, in: Random Graph 83, M. Karońki, A. Rucińki ed.), Amterdam, North Holland, 1985, pp [10] B. Bollobá, Random graph, London, Academic Pre,