Edge Cover Time for Regular Graphs

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1 Jounal of Intege Sequences, Vol. 11 (28, Aticle Edge Cove Time fo Regula Gahs Robeto Tauaso Diatimento di Matematica Univesità di Roma To Vegata via della Riceca Scientifica 133 Roma Italy tauaso@mat.unioma2.it Abstact Conside the following stochastic ocess on a gah: initially all vetices ae uncoveed and at each ste cove the two vetices of a andom edge. What is the exected numbe of stes equied to cove all vetices of the gah? In this note we show that the mean cove time fo a egula gah of N vetices is asymtotically (N log N/2. Moeove, we comute the exact mean cove time fo some egula gahs via geneating functions. 1 Intoduction The classical couon collecto s oblem can be extended in many ways. In some vaiants that can be found in the liteatue (see, fo examle, [1, 2, 3, 4] the objects to be collected ae the vetices of a gah. Thee ae vaious inteesting collection ocesses (e.g., a andom walk though the gah, but the following one does not seem to have been consideed much. Let G be a connected gah with N 2 vetices and M 1 edges (no loos. An edge coveing of G is a set of edges so that evey vetex of G is adjacent to (o coveed by at least one edge in this set. Initially all vetices of the gah ae uncoveed and at each ste we ick a andom edge among all edges and we cove its two vetices. Let τ G be the edge cove time, i.e., the andom vaiable that counts the numbe of stes equied to cove all vetices of G. What is its exected value E[τ G ]? Let C(G,k be the numbe of edge coveings of G with exactly k edges. Then the obability that at the n-th ste the whole gah is coveed is given by M { } n k! (n = C(G, k k M n. k=1 1

2 Since the following identity holds { } n k! = k then whee (n = Ĉ(G, = The obability geneating function is P(x = (nx n = n=1 k ( k ( 1 k n, M k= M Ĉ(G, ( n Ĉ(G, M M ( k ( 1 k C(G,k. n=1 ( x M M 1 n = x Ĉ(G, M x + because Ĉ(G,M = C(G,M = 1. In ode to comute E[τ G] we define Then Q(x = ((n (n 1x n + (1x = P(x(1 x. n=2 x 1 x Q (x = P (x(1 x P(x ( M 1 M 1 M = Ĉ(G, (M x + 1 x (1 x Ĉ(G, 2 (1 x 2 M x x 1 x ( M 1 M 1 M x = Ĉ(G, (1 x Ĉ(G, (M x 2 M x + 1. Finally we ae able to exess the answe in finite tems (see [6] and we obtain E[τ G ] = Q (1 = 1 M 1 Ĉ(G, M. (1 In the next sections we will aly the above fomula to seveal kind of gahs, afte the geneating function whose coefficients give C(G, k has been detemined. We decided to conside only egula gahs, so that no vetex is ivileged with esect to the othes. Befoe we stat, we would like to establish some bounds fo E[τ G ] when G is a geneic d-egula gah with N vetices (and dn/2 edges. Since the gah is egula, at each ste evey vetex has the same obability to be coveed. Hence if we assume that only one vetex of the chosen edge is coveed, then the modified ocess is just the classical couon collecto s oblem, and theefoe its mean cove time NH N is geate than E[τ G ]. A moe ecise asymtotic bound is given by the following theoem, which uses the obabilistic method (see, fo examle, [2]. 2

3 Theoem 1.1. Let G be a d-egula gah with N vetices and let τ G its cove time. Then fo any α > ( τ G (N log N/2 P N α 1 2e 2α + o(1. (2 Moeove, E[τ G ] (N log N/2. (3 Poof. Let A(v be the event such that the vetex v is not coveed afte f(n = N(log N + a/2 stes with a R. Since the obability that the vetex v is coveed at any ste is = d/(nd/2 = 2/N, it follows that P(A(v = (1 k 1 = (1 f(n = e a N + o(1/n. k>f(n If v w and v w is not an edge, then the obability that v o w ae coveed at any ste is = 2d/(Nd/2 = 4/N. Hence P(A(v A(w = (1 f(n = e 2a N 2 + o(1/n2. On the othe hand, if v w and v w is an edge then the obability that v o w ae coveed at any ste is = (2d 1/(Nd/2 = 4/N 2/(Nd then P(A(v A(w = (1 f(n = e a(2 1/d + o(1/n 2 1/d. N 2 1/d Let X(v be the indicato fo the event A(v and let X = v X(v be the numbe of vetices v such that A(v occus. We have the following estimates: E[X(v] = 1 P(A(v = e a N + o(1/n, Va[X(v] = E[(X(v] 2 E[X(v] 2 = e a N + o(1/n, Cov[X(v,X(w] = E[X(v X(w] E[X(v] E[X(w] = P(A(v A(w e 2a N + 2 o(1/n2 o(1/n 2, if v w is not an edge; = e a(2 1/d + o(1/n 2 1/d, if v w is not an edge. N 2 1/d 3

4 Theefoe E[X] = v E[X(v] = e a + o(1, Va[X] = Va[X(v] + Cov[X(v,X(w] v v w ( e = e a a(2 1/d + o(1 + Nd + o(1/n 2 1/d N 2 1/d +N(N 1 do(1/n 2 = e a + o(1. So we can find an exlicit ue and lowe bound fo P(X =, that is, the the obability that the cove time τ G is less than N(log N + a/2. By Chebyshev s inequality, P(X = P( X E[X] E[X] Va[X] (E[X] 2 = e a + o(1 e 2a + o(1 = ea + o(1. (4 On the othe hand P(X = = 1 P(X > 1 E[X] = 1 e a + o(1. (5 Let α >. By (4, if a = 2α then By (5, if a = 2M then Finally P(τ G (N log N/2 < αn e 2α + o(1. P(τ G (N log N/2 > αn = 1 P(τ G (N log N/2 < αn e 2α + o(1. P( τ G (N log N/2 < αn = 1 P(τ G (N log N/2 < αn P(τ G (N log N/2 > αn 1 2e 2α + o(1 and Eq. (2 has been oved. Eq. (3 follows diectly fom (2. 2 The cycle gah C n The cycle C n is a 2-egula gah with N = n vetices laced aound a cicle and M = n edges. In ode to comute the numbe of ways C(C n,k,v such that k edges cove v vetices of C n, we choose one of the n vetices and, fom thee, we lace clockwise the v k connected comonents. Let x i 1 be numbe of edges of the ith-comonent then these numbes solve the equation x 1 + x x v k = k. Let y i 1 be numbe of edges of the ga between ith-comonent and the next one then these numbes solve equation y 1 + y y v k = n k. 4

5 y 3 x 3 x 1 y 2 x 2 y 1 Hence n times the numbe of the all ositive integal solutions of the evious equations gives v k times (the fist comonent is labeled the numbe C(C n,k,v. Theefoe C(C n,k,v = n ( ( k 1 n k 1 = n ( ( k n k 1 v k v k 1 v k 1 k v k v k 1 and the numbe of edge coveings with k edges is C(C n,k = C(C n,k,n = n ( k = [x n y k 2 xy ] k n k 1 yx yx 2. The sequence is tiangula with esect to the double index (n,k since C(C n,k can be consideed zeo when k > n, and it aeas in Sloane s Encycloedia [7] as A It is inteesting to note that the total numbe of edge coveings of C n is the n-th Lucas numbe (A32 ( n k C(C n = C(C n,k = = L n. k n k k=1 This is not a eal suise because the edge coveings of C n ae in bijective coeondence with the monome-dime tilings (no ovelaing of C n : elace any vetex coveed by two edges with a monome and then fill the est with dimes. k=1 The following identities ae well known (see, fo examle, [5] and we include the oofs hee fo comleteness. 5

6 Lemma 2.1. i Fo any ositive intege n ii Fo any ositive integes n and Poof. As egads identity (6 and 1 (1 x n 1 1 x 1 Now identity (7, 1 ( ( 1 n 1 1 n = nh n. (6 = dx = (1 x n 1 1 x x 1 (1 x n 1 dx = and on the othe hand ( ( 1 n 1 = 1 ( n 1 1 n 1 ( 1 ( n 1 x 1 dx = 1 t n 1 1 dx = t 1 dx = 1 x 1 = ( 1 ( n 1 1 n 2 (7 ( ( 1 n 1 t dt = H n 1. = x dx = =, ( ( 1 n 1, 1 x 1 (1 x n 1 dx = B( 1,n 1 = ( 1!(n 1! (n 1! = 1 ( n 1. We ae now able to find an exlicit fomula fo the mean cove time fo the cycle gah. Theoem 2.2. n/2 E[τ Cn ] = nh n n =1 ( n ( n

7 Poof. By (1 E[τ Cn ] = 1 Ĉ(C n, n = 1 Ĉ(C n,n n n ( ( k n k = 1 ( 1 k n+ n k n k k=n 1 ( ( k 1 k = 1 n ( 1 k n+ n 1 n k k=n 1 ( ( n 1 n = 1 n ( 1 = ( ( 1 n 1 ( n ( ( 1 n 1 = 1 n n. Theefoe, by the evious lemma E[τ Cn ] = nh n n =1 ( n ( n 1 =1 = nh n n = n/2 =1 ( n ( n 1. The exact values of E[τ Cn ] fo N = n = 2, 3, 4, 5, 6, 7, 8 ae 1, 5 2, 11 3, 31 6, 67 1, 167 2, The cyclic ladde C n K 2 The cyclic ladde is a 3-egula gah obtained by taking the gah catesian oduct of the cycle gah C n and the comlete gah K 2. It has N = 2n vetice (n on the oute cicle and n in the inne cicle and M = 3n edges (n on each cicle and the n ungs. 7

8 The numbe of coveings of C n K 2 without ungs is L 2 n because the inne and the oute cicles ae coveed indeendently. Assume that the coveing has 1 ungs then we label the fist one and we let x i be the numbe of edges (on one cicle between the i-th ung and the next one. Since the numbe of coveings of a linea gah with x i + 2 vetices with the end vetices aleady coveed is the Fibonacci numbe F xi +3 (just the numbe of monome-dime tilings of the (x i + 2-sti then the numbe of coveings with 1 ungs is given by the -convolution (n/ Fx 2 i +3. Theefoe C(C n K 2 = L 2 n + x 1 + +x =n (n/ i=1 x 1 + +x =n Now it is easy to find the geneating function: since h(x = L 2 nx n 4 7x x 2 = and g(x = (1 + x(1 3x + x 2 n= it follows that ( (xg(x f(x = h(x + (xd Fx 2 i +3. i=1 Fn+3x 2 n = n= ( ( = h(x + (xd log 4 + x x 2 (1 + x(1 3x + x 2, 1 1 xg(x 4 15x 18x 2 x 3 = (1 + x(1 6x 3x 2 + 2x 3 = 4 + 5x + 43x x x x x x 7 + o(x 7 and C(C n K 2 = [x n ]f(x is the sequence A Letting 4 (4y + 3y 2 x y 3 x 2 h(x,y = 1 (y + y 2 x (y 2 + y 3 x 2 + y 3 x 3 and g(x = 1 + 2y + y2 + ( y + y 2 + y 3 x y 3 x 2, 1 (y + y 2 x (y 2 + y 3 x 2 + y 3 x 3 by a simila agument, we can show that C(C n K 2,k = [x n y k ]f(x,y whee ( f(x,y = h(x,y + x ( ( 1 log x 1 xyg(x,y = 4 (3y + 9y2 + 3y 3 x (4y 2 + 1y 3 + 4y 4 x 2 + (y 3 y 4 y 5 x 3. (1 + xy(1 (2y + 3y 2 + y 3 x (2y 3 + y 4 x 2 + (y 3 + y 4 x 3 By (1, the exact values of E[τ Cn K 2 ] fo N = 2n = 4, 6, 8, 1 ae 18 5, , ,

9 4 K n and K n,n Thee ae two othe imotant egula gahs whose edge coveings ae counted by sequences contained in the Sloane s Encycloedia [7]: the comlete gah K n and the comlete biatite gah K n,n. All the fomulas can be veified by alying the inclusion-exclusion incile. The tiangula sequence C(K n,k fo k M = ( n 2 is A54548 and C(Kn is A6129: C(K n,k = ( n ( 1 j j j= (( n j 2 k and C(K n = By (1, the exact values of E[τ Kn ] fo N = n = 2, 3, 4, 5, 6 ae 1, 5 2, 19 5, , ( n ( 1 j 2 (n j 2. j The tiangula sequence C(K n,n,k fo k M = n 2 is A55599 and C(K n,n is A48291 and C(K n,n,k = j= ( ( ( n n (n j(n i ( 1 i ( 1 j i j k i= C(K n,n = j= j= ( n ( 1 j (2 n j 1 n. j By (1, the exact values of E[τ Kn,n ] fo N = 2n = 2, 4, 6, 8, 1 ae 1, 11 3, , , Refeences [1] M. Adle, E. Halein, R. Ka and V. Vaziani, A stochastic ocess on the hyecube with alications to ee-to-ee netwoks, Poc. STOC 23, [2] N. Alon and J. H. Sence, The Pobabilistic Method, 2nd Edition, Wiley, 2. [3] C. Cooe and A. Fieze, The cove time of andom egula gahs, SIAM J. Discete Math., 18 (25, [4] N. Dimitov and C. Plaxton, Otimal cove time fo a gah-based couon collecto ocess, Poc. ICALP 25, [5] H. W. Gould, Combinatoial Identities, Mogantown, West Viginia, [6] A. N. Myes and H. Wilf, Some new asects of the couon collecto s oblem, SIAM J. Discete Math., 17 (23,

10 [7] N. J. A. Sloane, The On-Line Encycloedia of Intege Sequences, ublished electonically at htt:// njas/sequences/. 2 Mathematics Subject Classification: Pimay 6C5. Keywods: couon collecto s oblem, gah, edge cove. (Concened with sequences A32, A6129, A48291, A54548, A55599, A113214, and A Received Octobe 23 26; evised vesion eceived May 24 28; Setembe Published in Jounal of Intege Sequences, Octobe Retun to Jounal of Intege Sequences home age. 1