ALAN FRIEZE, CHARALAMPOS E. TSOURAKAKIS

Transcription

1 RAINBOW CONNECTIVITY OF G,p) AT THE CONNECTIVITY THRESHOLD ALAN FRIEZE, CHARALAMPOS E. TSOURAKAKIS Abstract. A edge colored graph G is raibow edge coected if ay two vertices are coected by a path whose edges have distict colors. The raibow coectivity of a coected graph G, deoted by rcg), is the smallest umber of colors that are eeded i order to make G raibow coected. I this work we study the raibow coectivity of the biomial graph G = G,p) at the coectivity threshold p = log+ω where ω = ω) ad ω = olog). We prove that the raibow coectivity of G satisfies rcg) max{z 1,diameterG)} with high probability whp). Here Z 1 is the umber of vertices i G whose degree equals 1 ad the diameter of G is asymptotically equal to log loglog, whp. 1. Itroductio Coectivity is a fudametal graph theoretic property. Recetly, the cocept of raibow coectivity was itroduced by Chartrad et al. i [5]. A edge colored graph G is raibow edge coected if ay two vertices are coected by a path whose edges have distict colors. The raibow coectivity rcg) of a coected graph G is the smallest umber of colors that are eeded i order to make G raibow edge coected. Notice, that by defiitio a raibow edge coected graph is also coected ad furthermore ay coected graph has a trivial edge colorig that makes it raibow edge coected, sice oe may color the edges of a give spaig tree with distict colors. Other basic facts established i [5] are that rcg) = 1 if ad oly if G is a clique ad rcg) = VG) 1 if ad oly if G is a tree. Besides its theoretical iterest, raibow coectivity is also of iterest i applied settigs, such as securig sesitive iformatio [9] trasfer ad etworkig [4]. For istace, cosider the followig settig i etworkig [4]: we wat to route messages i a cellular etwork such that each lik o the route betwee two vertices is assiged with a distict chael. The, the miimum umber of chaels to use is equal to the raibow-coectivity of the uderlyig etwork. The cocept of raibow coectivity has attracted the iterest of various researchers. Chartrad et al. [5] determie the raibow coectivity of several special classes of graphs, icludig multipartite graphs. Caro et al. [3] prove that for a coected graph G with vertices ad miimum degree δ, the raibow coectivity satisfies rcg) logδ 1 + fδ)), where fδ) teds to zero as δ δ icreases. The followig simpler boud was also proved i [3], rcg) 4log+3. Krivelevich δ ad Yuster more recetly [8] removed the logarithmic factor from the Caro et al. [3] upper boud. Specifically they proved that rcg) 20. It is worth oticig that due to a costructio of a graph δ 3δ with miimum degree δ ad diameter δ+7 by Caro et al. [3], the best upper boud oe ca +1 δ+1 hope for is rcg) 3. δ As Caro et al. poit out, the radom graph settig poses several itriguig questios. Specifically, let G = G,p) deote the biomial radom graph o vertices with edge probability p [6]. Caro et al. [3] proved that p = log/ is the sharp threshold for the property rcg,p)) 2. He ad Liag [7] studied further the raibow coectivity of radom graphs. Specifically, they obtai the sharpthresholdforthepropertyrcg) dwherediscostat. Forfurtherresultsadrefereceswe 1

2 2 ALAN FRIEZE, CHARALAMPOS E. TSOURAKAKIS refer the iterested reader to the recet survey of Li ad Su [9]. I this work we look at the raibow coectivity of the biomial graph at the coectivity thresold p = log+ω where ω = olog). This rage of values for p poses problems that caot be tackled with the techiques developed i the aforemetioed work. Let L = log loglog. We establish the followig theorem: Theorem 1. Let G = G,p),p = log+ω, ω,ω = olog). Also, let Z 1 be the umber of vertices of degree 1 i G. The, with high probabilitywhp) 1 rcg) max{z 1,L}, where A B deotes A = 1+o1))B. All logarithms, uless explicitly stated, are atural. 2. Proofs Observe first that rcg) max{z 1,diameterG)}. First of all, each edge icidet to a vertex of degree oe must have a distict color. Just cosider a path joiig two such vertices. Secodly, if the shortest distace betwee two vertices is l the we eed at least l colors. Next observe that whp the diameter D is asymptotically equal to L, see for example [2]. We break the proof Theorem 1 ito several lemmas. Let a vertex be large if degx) log/100 ad small otherwise. Lemma 1. Whp, there do ot exist two small vertices of degree withi distace at most 3L/4. Proof. We defie the evet { Q = x,y [] : degx),degy) log/100 ad distx,y) 3L 4 The, Pr[Q] )3L/4 k 1 p k 2 3L/4 k=1 3L/4 k=1 3L/4 k=1 k=1 2log) 2 k log /100 i=0 log /100 ) 1 k p i 1 p) 1 k i 2 }. )p log/100 1 p) 1 log/100 ) 2 2log) k 2100e 1+o1) ) log/100 1+o1)) 2 2log) k log) 3L/ A evet A holds with high probability whp) if lim + Pr[A ] = 1.

3 RAINBOW CONNECTIVITY OF G, p) AT THE CONNECTIVITY THRESHOLD 3 Figure 1. Structure of Lemma 3. Lemma 2 is critical for Lemma 3 i order to prove that almost all edges explored by umerous Breadth First Search BFS) processes will be useful i creatig appropriate tree structures. We use the otatio e[s] for the umber of edges iduced by a give set of vertices S. Notice that if a set S satisfies e[s] s+t where t 1, the iduced subgraph G[S] has at least t+1 cycles. Lemma 2. Whp there does ot exist a subset S [], such that S αtl where 0 < α < 1 ad t Z + is a costat ad e[s] S +t. Proof. For coveiece, let s = S be the cardiality of the set S.The, Pr[ S : s αtl ad e[s] s+t] ) s ) 2) p s+t s s+t s αtl s αtl e ) s es 2 p s 2s+t) ) s+t ) t eslog e 2+o1) log) s s αtl eαtlog 2 )) t αtl e 2+o1) log) αl loglog 1 < 1 α o1))t. Lemma 3 shows the existece of the subgraph G x,y described ext ad show i Figure 1 for a give pair of vertices x,y. We first deal with paths betwee large vertices. Now let ǫ = ǫ) = o1) be such that ǫloglog ad let k = ǫl. log 1/ǫ Lemma 3. For all pairs of large vertices x,y [] there exists a subgraph G x,y V x,y,e x,y ) of G as show i figure 1, whp. The subgraph cosists of two isomorphic disjoit trees T x,t y rooted at x,y each of depth k. T x ad T y both have a brachig factor of log/101. If the leaves of T x are

4 4 ALAN FRIEZE, CHARALAMPOS E. TSOURAKAKIS Figure 2. Subgraph foud i the proof of Lemma 3. x 1,x 2,...,x τ,τ 4ǫ/5 the y i = fx i ) where f is a atural isomporphism. Betwee each pair of leaves x i,y i ),i = 1,2,...,m there is a path P i of legth 1+2ǫ)L. The paths P i,i = 1,2,...,m are edge disjoit. Proof. Because we have to do this for all pairs x,y, we ote without further commet that likely resp. ulikely) evets will be show to occur with probability 1 o 2 ) resp. o 2 )). To fid the subgraph show i Figure 1 we grow tree structures as show i Figure 2. Specifically, we first grow a tree from x usig BFS util it reaches depth k. The, we grow a tree startig from y agai usig BFS util it reaches depth k. Fially, we grow trees from the leaves of T x ad T y usig BFS for depth γ = 1 +ǫ)l. Now we aalyze these processes. Sice the argumet is the same we 2 explai it i detail for T x ad we outlie the differeces for the other trees. We use the otatio D r) i for the umber of vertices at depth i of the BFS tree rooted at r. First we grow T x. As we grow the tree via BFS from a vertex v at depth i to vertices at depth i + 1 certai bad edges from v may poit to vertices already i T x. Lemma 2 with t = 10,α = ǫ shows with probability at least 1 ) 1 10 there exists o subset of vertices S of cardiality at 1 2ǫ most 10ǫL that cotais S +10 edges. Cosequetly, there ca be at most 10 bad edges emaatig from v. Furthermore, Lemma 1 implies that there exists at most oe vertex of degree less tha log 100 at each level whp. Hece, we obtai the recursio Therefore the umber of leaves satisfies ) D x) log i D x) i 1) log 101 Dx) i. 1) D x) k We ca make the brachig factor exactly log by pruig. 101 With a similar argumet ) ǫl log 4ǫ/5. 2) 101 D y) k 4 5 ǫ. 3)

5 RAINBOW CONNECTIVITY OF G, p) AT THE CONNECTIVITY THRESHOLD 5 The oly differece is that ow we also say a edge is bad if the other edpoit is i T x. This immediately gives ) D y) log i D x) i 1) log 101 Dx) i ad the required coclusio 3). Similarly, from each leaf x i T x ad y i T y we grow trees T xi, T yi of depth γ = 1 +ǫ) L usig 2 the same procedure ad argumets as above. To argue for few bad edges we apply Lemma 2 with parameters α = 2/3 ad t = 10 to show there are few edges from the vertex v beig explored to vertices i ay of the trees already costructed. The umbers of leaves of each T xi ow satisfies D x i) γ log ) γ log ǫ Similarly for D y i) γ. Observe ext that BFS does ot coditio the edges betwee the leaves X i,y i of the trees T xi ad T yi. I.e., we do ot eed to look at these edges i order to carry out our costructio. O the other had we have coditioed o the occurece of certai evets to imply a certai growth rate. We hadle this techicality as follows. We go through the above costructio ad halt if ever we fid that we caot expad by the required amout. Let A be the evet that we do ot halt the costructio. We have Pr[A] = 1 o1) ad we ca upper boud coditioal probabilities by iflatig ucoditioed probabilities by 1 + o1). So, Pr[ i : ex i,y i ) = 0 A] 2 4ǫ 5 1 p) 1+8ǫ 5 ǫ. Let C = 1+5ǫ)L be the umber of available colors. We color the edges of G radomly. Specifically, weshowthattheprobabilityofhavigaraibowpathbetweethemithesubgraphg x,y offigure1 is at least Lemma 4. Color each edge of G usig oe color at radom from C available. The, the probability of havig at least oe raibow path betwee two fixed large vertices x,y [] is at least Proof. We show that the subgraph G x,y cotais such a path. We break our proof ito two steps: Before we proceed, we provide certai ecessary defiitios. Thik of the colorig process as a evolutioary process that colors edges by startig from the two roots x,fx) = y util it reaches the leaves. I the followig, we call a vertex u of T x T y ) alive if the path Px,u) Py,u)) from x y) to u is raibow, i.e., the edges have received distict colors. We call a pair of vertices {u,fu)} alive, u T x,fu) T y if u,fu) are both alive ad the paths Px,u),Py,fu)) share o color. Defie A j = {u,fu)) : u,fu)) is alive ad depthu) = j} for j = 1,..,k. Step 1: Existece of at least 4 5 ǫ livig pairs of leaves Assume the pair of vertices {u,fu)} is alive where u T x,fu) T y. It is worth oticig that u,fu) have the same depth i their trees. We are iterested i the umber of pairs of childre {u i,fu i )} i=1,..,log/101 that will be alive after colorig the edges from depthu) to depthu) + 1. A livig pair {u i,fu i )} by defiitio has the followig properties: edges u,u i ) ET x ) ad fu),fu i )) ET y ) receive two distict colors, which are differet from the set of colors used i paths Px,u) ad Py,fu)). Notice the latter set of colors has cardiality 2 depthu) 2k.

6 6 ALAN FRIEZE, CHARALAMPOS E. TSOURAKAKIS Figure 3. Figure shows log -ary trees T 101 x,t y. The two roots are show respectively at the ceter of the trees. I our thikig of the radom colorig as a evolutioary process, the gree edges icidet to x survive with probability 1, the red edges icidet to y with probability 1 1 ad all the other edges with probability p C 0 = 1 2k C where k is the depth of both trees ad C the umber of available colors. Our aalysis i Lemma 3 usig these probabilities gives a lower boud o the umber of alive pairs of leaves after colorig T x,t y from the root to the leaves respectively. ) 2 Let A j be the umber of livig pairs at depth j. We first boud the size of A 1. [ Pr A 1 log ] log/ log/200 = O 200 C) Ωloglog) ). 4) For j > 1 we see that the radom variable equal to the umber of livig pairs of childre of u,fu)) stochastically domiates the radom variable X Biblog,p 0 ), where p 0 = ) 1 2k 2 C = 1+3ǫ ǫ) We have usig the Cheroff bouds for 2 j k, Pr [ A j < log 200 ) j p j 1 log 0 A j { exp 1 2 ) j 1 p j 2 0 ] ) 2 99 log ) ad 5) justify assumig that that A k ) log k ǫ. log 200 ) j 1 p j 0 Step 2: Existece of raibow paths betwee x,y i G x,y Assumig that there are 4ǫ/5 livig pairs of leaves x i,y i ) for vertices x,y, Prx,y are ot raibow coected) 1 2γ 1 i=0 1 } = O Ωloglog) ). 5) ) ) 4ǫ/5 2k +i. C

7 RAINBOW CONNECTIVITY OF G, p) AT THE CONNECTIVITY THRESHOLD 7 a) b) But So 2γ 1 i=0 Figure 4. Takig care of small vertices. 1 ) 2k +i 1 C Prx,y are ot raibow coected) exp This completes the proof of Lemma 4. { ) 2γ ) 2γ 2k +2γ ǫ =. C 1+5ǫ ) } 2γ ǫ 4ǫ/5 1+5ǫ We ow fiish the proof of Theorem 1 i.e. take care of small vertices. = exp { 4ǫ/5 Olog1/ǫ)/loglog)}. Proof. We showed i Lemma 4 that whp for ay two large vertices, our colorig results i a raibow path joiig them. We divide the small vertices ito two sets: vertices of degree 1, V 1 ad the vertices of degree at least 2, V 2. Suppose that our colors are 1,2,...,C ad V 1 = {v 1,v 2,...,v r }. We begi by givig the edge icidet with v i the color i. The we slightly modify the argumet i Lemma 4. If x is the eighbor of v i V 1 the color i caot be used i Steps 1 ad 2 of that procedure. I terms of aalyis this replaces C by C 1) C 2) if y is also a eighbor of V 1 ) ad the argumet is essetially uchaged. The set V 2 is treated by usig oly two extra colors. Assume that Red ad Blue have ot bee used i our colorig. The we use Red ad Blue to color two of the edges icidet to a vertex u V 2 the remaiig edges are colored arbitrarily). This is show i Figure 4b. Suppose that V 2 = {w 1,w 2,...,w s }. The if we wat a raibow path joiig w i,w j where i < j the we use the red edge to go to its eighbor w i. The we take the already costructed raibow path to w j, the eighbor of w j via a blue edge. The we ca cotiue to w j. 3. Coclusio I this work we studied the problem of determiig the raibow coectivity of G = G,p) at the coectivity threshold. We have determied rcg) asymptotically. It is reasoable to cojecture that this could be tighteed:

8 8 ALAN FRIEZE, CHARALAMPOS E. TSOURAKAKIS Cojecture: Whp, rcg) = max{z 1,diameterG,p))}. Refereces [1] Aath, P., Nasre, M., Sarpatwar, K.: Raibow Coectivity: Hardess ad Tractability. IARCS Aual Coferece o Foudatios of Software Techology ad Theoretical Computer Sciece FSTTCS), pp ) [2] Bollobás, B.: Radom Graphs. Cambridge Uiversity Press 2001) [3] Caro, Y., Lev, A., Roditty, Y., Tuza, Z., Yuster, R.: O raibow coectio. Electroic Joural of Combiatorics, Vol ) [4] Chakrabory, S., Fischer, E., Matsliah, A., Yuster, R.: Hardess ad Algorithms for Raibow Coectio. Joural of Combiatorial Optimizatio, Vol. 213) 2011) [5] Chartrad, G., Johs, G.L., McKeo, K.A., Zhag, P.: Raibow coectio i graphs. Mathematica Bohemica, Vol. 1331), pp ) [6] Erdös, P., Réyi, A.: O Radom Graphs I. Publicatioes Mathematicae, Vol. 6, pp ) [7] He, J, Liag, H.: O raibow-k-coectivity of radom graphs. Arxiv v1, available at ) [8] Krivelevich, M., Yuster, R.: The raibow coectio of a graph is at most) reciprocal to its miimum degree. Joural of Graph Theory, Vol. 633), pp ) [9] Li, X., Su, Y.: Raibow coectios of graphs - A survey. Arxiv v2, available at ) Departmet of Mathematical Scieces, Caregie Mello Uiversity, 5000 Forbes Av., 15213, Pittsburgh, PA, U.S.A address: af1p@math.radom.cmu.edu,ctsourak@math.cmu.edu