On decompositions of complete multipartite graphs into the union of two even cycles

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1 On decompositions of complete multipatite gaphs into the union of two even cycles A. Su, J. Buchanan, R. C. Bunge, S. I. El-Zanati, E. Pelttai, G. Rasmuson, E. Spaks, S. Tagais Depatment of Mathematics Illinois State Univesity Nomal, IL U.S.A. Abstact Fo positive integes c and d, let K c d denote the complete multipatite gaph with c pats, each containing d vetices. Let G with n edges be the union of two vetex-disjoint even cycles. We use gaph labelings to show that thee exists a cyclic G-decomposition of K n+1) t, K n/+1) t, K 5 n/)t, and of K nt fo evey positive intege t. If n 0 mod ), then thee also exists a cyclic G-decomposition of K n+1) t, K n/+1) 8t, K 9 n/)t, and of K 3 nt fo evey positive intege t. 1 Intoduction If a and b ae integes we denote {a, a + 1,..., b} by [a, b] if a > b, [a, b] = ). Let N 0 denote the set of nonnegative integes and Z n the goup of integes modulo n. Fo a gaph G, let V G) and EG) denote the vetex set of G and the edge set of G, espectively. Let K k denote the complete gaph on k vetices. Let V K k ) = Z k and let G be a subgaph of K k. The length of an edge {i, j} EG) is defined as min{ i j, k i j }. By clicking G, we mean applying the isomophism i i + 1 to V G). Let H and G be gaphs such that G is a subgaph of H. A G-decomposition of H is a set Γ = {G 1, G,..., G t } of paiwise edge-disjoint subgaphs of H each of which is isomophic to G and such that EH) = t EG i). If H is K k, a G-decomposition Γ of H is cyclic if clicking is an automophism of Γ. The Reseach suppoted by National Science Foundation Gant No. A

2 decomposition is puely cyclic if it is cyclic and Γ = V H). If G is a gaph and is a positive intege, G denotes the vetex disjoint union of copies of G. The study of gaph decompositions, also known as the study of gaph designs o G-designs, is a popula aea of eseach. In paticula, decompositions of complete gaphs into cycles have attacted a geat deal of attention. Fo elatively ecent suveys on gaph decompositions, we diect the eade to [] and [5]. A popula method fo obtaining gaph decompositions is via gaph labelings. Fo any gaph G, a one-to-one function f : V G) N 0 is called a labeling o a valuation) of G. In [1], Rosa intoduced a hieachy of labelings. Let G be a gaph with n edges and no isolated vetices and let f be a labeling of G. Let fv G)) = {fu) : u V G)}. Define a function f : EG) Z + by fe) = fu) fv), whee e = {u, v} EG). We will efe to fe) as the label of e. Let feg)) = { fe) : e EG)}. Conside the following conditions: l1) fv G)) [0, n], l) fv G)) [0, n], l3) feg)) = {x 1, x,..., x n }, whee fo each i [1, n] eithe x i = i o x i = n + 1 i, l) feg)) = [1, n]. If in addition G is bipatite with vetex bipatition {A, B}, conside also l5) fo each {a, b} EG) with a A and b B, we have fa) < fb), l6) thee exists an intege λ such that fa) λ fo all a A and fb) > λ fo all b B. Then a labeling satisfying the conditions: l1), l3) l1), l) is called a ρ-labeling; is called a σ-labeling; l), l) is called a β-labeling. A β-labeling is necessaily a σ-labeling which in tun is a ρ-labeling. Suppose G is bipatite. If a ρ-, σ-, o β-labeling of G satisfies condition l5), then the labeling is odeed and is denoted by ρ +, σ +, o β +, espectively. If in addition l6) is satisfied, the labeling is unifomly odeed and is denoted by ρ ++, σ ++, o β ++, espectively. A β-labeling is bette known as a gaceful labeling and a unifomly odeed β-labeling is an α-labeling as intoduced in [1]. Labelings of the

3 types above ae called Rosa-type labelings because of Rosa s oiginal aticle [1] on the topic see [10] fo a compehensive suvey of Rosa-type labelings). A dynamic suvey on geneal gaph labelings is maintained by Gallian [11]. Labelings ae citical to the study of cyclic gaph decompositions as seen in the following two esults fom [1] and [9], espectively. Theoem 1. Let G be a gaph with n edges. Thee exists a puely cyclic G-decomposition of K n+1 if and only if G has a ρ-labeling. Theoem. Let G be a gaph with n edges that admits a ρ + -labeling. Then thee exists a cyclic G-decomposition of K nx+1 fo all positive integes x. d-modula labelings and decompositions of K c dt Fo positive integes c and d, let K c d denote the complete multipatite gaph with c pats, each containing d vetices. Note that K c d has cd vetices and c ) d edges. We can conside K c d as a subgaph of the complete gaph K cd, with V K c d ) = Z cd and EK c d ) = { {u, v} : u, v Z cd, u v mod c) }, that is, the c pats of K c d ae the conguence classes of Z cd modulo c. Note that K c d has pecisely the edges of K cd whose lengths ae not multiples of c. Let G be a gaph and let {G 1, G,..., G t } be a G-decomposition of K c d with V K c d ) = Z cd as defined above). If clicking pemutes the gaphs in the decomposition, then we say that it is a cyclic G-decomposition of K c d, and if clicking G 1 cd 1 times poduces each gaph in the decomposition exactly once, then we say the decomposition is puely cyclic. In the latte case if G has n edges, we must have c ) d = ncd, and so c = n/d + 1. Suppose that G is a gaph with n edges and d is a positive intege such that d divides n. Set c = n/d + 1, so that cd = n + d. By a d-modula ρ-labeling of G we mean a one-to-one function f : V G) [0, cd 1] such that { { } } [ min fu) fv), cd fu) fv) : {u, v} EG) = 1, cd ] \cz. In othe wods, a d-modula ρ-labeling of a gaph with n edges has evey edge length in K n+d exactly once except fo any multiples of n/d + 1. Figue 1 shows an example of a 3-modula ρ-labeling of a 6-cycle. As a subgaph of K 15, the edge length 5 is missing. Thus this C 6 has one edge of each length in K 5 3 and clicking it 1 times would poduce a puely cyclic C 6 -decomposition of K 5 3. Thus fom the definition of d-modula ρ-labelings, it is staightfowad to see that the following holds. 3

4 Theoem 3. If the gaph G with n edges admits a d-modula ρ-labeling and c = n/d + 1, then K c d has a puely cyclic G-decomposition. We obseve that a ρ-labeling of G is necessaily a 1-modula ρ-labeling. Moeove, a σ-labeling of G is necessaily a -modula ρ-labeling. We also note the following. Theoem. Let G be a bipatite gaph with n edges. If G admits a ρ + - labeling, then G admits a n-modula ρ-labeling. Poof. Let {A, B} be a bipatition of V G) and let f be a ρ + -labeling of G such that fa) < fb) fo evey {a, b} EG) with a A and b B. Define a labeling g : V G) [0, n 1] by ga) = fa) fo a A and gb) = fb) 1 fo b B. It is easy to veify that g is a n-modula ρ-labeling of G. Next we note that if evey vetex of a gaph G has even degee, then in a d-modula labeling of G, the numbe of edges with an odd label must be even. This is known as the paity condition. Lemma 5. Let G be a gaph with all even degees and let f be a d-modula labeling of G. Let O = {e EG): fe) is odd}. Then O is even. Poof. Fo e = {u, v} EG), eithe fe) = fu) fv) o fe) = fv) fu). Let S = fe). e EG) Let v V G). Since degv) is even, the sum of the numbe of occuences of fv) and of fv) in S is even. Theefoe S is even and hence O must be even. The concept of a d-modula ρ-labeling elates vey closely to the concepts of diffeence families and diffeence matices developed by Buatti and seveal co-authos ove the last seveal yeas. See fo example, Buatti [6], Buatti and Gionfiddo [7], and Buatti and Pasotti [8]. Anothe elated concept is that of a d-gaceful labeling as intoduced by Pasotti in [13]. Rathe than define these additional concepts hee, we state a poweful esult on d-modula ρ-labelings that can be obtained fom the main esult on gaph decompositions with the use of diffeence matices in [8]. Theoem 6. If a z-patite gaph G with n edges has a d-modula ρ-labeling and c = n/d+1, then K c td has a cyclic G-decomposition fo evey positive intege t such that gcdt, z 1)!) = 1. Thus if G is bipatite, then we have the following coollay to Theoem 6. Coollay 7. If a bipatite gaph G with n edges has a d-modula ρ-labeling and c = n/d+1, then K c td has a cyclic G-decomposition fo evey positive intege t.

5 We illustate how the esult in Coollay 7 woks. Let {A, B} be a bipatition of V G) and let f be a d-modula ρ-labeling of G. Let A = {u 1, u,..., u } and B = {v 1, v,..., v s }. Let x be a positive intege. Fo 1 i x, let G i be a copy of G with bipatition A, B i ) whee B i = {v i,1, v i,,..., v i,s } and v i,j coesponds to v j in B. Let Gx) = G 1 G G x. Thus Gx) is bipatite with bipatition {A, B 1 B B x }. Define a labeling f of Gx) as follows: f a) = fa) fo each a A and f v i,j ) = fv j ) + i 1)n + d) fo 1 i x and 1 j s. It is easy to see that f is a d-modula ρ-labeling of Gx) and thus Theoem 3 applies. Figue 1 shows a 3-modula ρ-labeling of C 6 and the thee states fo a cyclic C 6 -decomposition of K 5 9 that can be obtained fom that 3-modula ρ-labeling of C = Figue 1: A 3-modula ρ-labeling of C 6 and thee states fo a cyclic C 6 - decomposition of K 5 9. In this aticle, we investigate the existence of d-modula ρ-labelings fo the gaph G consisting of the vetex-disjoint union of two even cycles. In light of Coollay 7, these labelings lead to cyclic G-decompositions of vaious infinite classes of complete multipatite gaphs. In [13], Pasotti poduces labelings of C k that lead to cyclic C k -decompositions of K k+1) n and of K k+1) 8n fo all positive integes k and n. She also poduces labelings that lead to cyclic C k -decompositions of K k+1) n fo all odd integes k 1 and all positive integes n. In [3], Benini and Pasotti efine the esults fom [13] to poduce labelings of C k that yield cyclic C k -decompositions of K k d +1) dn fo any positive integes k, n and any positive diviso d of k. Numeous othe authos have studied decompositions not necessaily cyclic ones) of complete multipatite gaphs into cycles. Paticula focus has been placed on C 3 -decompositions of complete multipatite gaphs. Such decompositions fall unde the umbella of the study of goup divisible designs see [1] fo a summay). The poblem of C k -decompositions of the complete bipatite gaph K m,n was settled completely by Sotteau in [15]. 5

6 3 Additional Notation We denote the diected path with vetices x 0, x 1,..., x k, whee x i is adjacent to x i+1, 0 i k 1, by x 0, x 1,..., x k ). The fist vetex of this path is x 0, the second vetex is x 1, and the last vetex is x k. If x 0, x 1,..., x k, ae distinct vetices, then the path x 0, x 1,..., x k, x 0 ) is necessaily a cycle on k + 1 vetices. If G 1 = x 0, x 1,..., x j ) and G = y 0, y 1,..., y k ) ae diected paths with x j = y 0, then by G 1 + G we mean the path x 0, x 1,..., x j, y 1, y,..., y k ). Let P k) be the path with k edges and k + 1 vetices 0, 1,..., k given by 0, k, 1, k 1,, k,..., k/ ). Note that the set of vetices of this gaph is A B, whee A = [0, k/ ], B = [ k/ + 1, k], and evey edge joins a vetex of A to one of B. Futhemoe, the set of labels of the edges of P k) is [1, k]. Now let a and b be nonnegative integes with a b and let us add a to all the vetices of A and b to all the vetices of B. We will denote the esulting gaph by P a, b, k). Note that this gaph has the following popeties. P1) P a, b, k) is a path with fist vetex a and second vetex b + k. Its last vetex is a + k/ if k is even and b + k + 1)/ if k is odd. P) Each edge of P a, b, k) joins a vetex of A = [a, k/ + a] to a lage vetex of B = [ k/ b, k + b]. P3) The set of edge labels of P a, b, k) is [b a + 1, b a + k]. Now conside the diected path Qk) obtained fom P k) eplacing each vetex i with k i. The new gaph is the path k, 0, k 1, 1,..., k k/ ). The set of vetices of Qk) is A B, whee A = k B = [0, k k/ 1] and B = k A = [k k/, k], and evey edge joins a vetex of A to one of B. The set of edge labels is still [1, k]. The last vetex of Qk) is k/ B if k is even and k 1)/ A if k is odd. We add a to the vetices of A and b to vetices of B, whee a and b ae integes, 0 a b. This gaph is k + b, a, k + b 1, a + 1,...) which we will denote by Qa, b, k). Note that this gaph has the following popeties. Q1) Qa, b, k) is a path with fist vetex k + b. Its last vetex is b + k/ if k is even and a + k 1)/ if k is odd. Q) Each edge of Qa, b, k) joins a vetex of A = [a, a + k k/ 1] to a lage vetex of B = [b + k k/, b + k]. Q3) The set of edge labels of Qa, b, k) is [b a + 1, b a + k]. 6

7 a) P 3, 8, 5) b) Q5, 7, 6) Figue : Examples of the path notations with an even numbe of edges. Main Results Lemma 8. A d-modula ρ-labeling of C C s exists fo 1 s and d {1,,, 8, + s, + s), + s), 8 + s)}. Poof. Let G = C C s whee, s 1. The cases d = 1, d =, and d = 8 +s) can be obtained fom the fact that such a G necessaily admits an α-labeling see [1]). Case 1: d =. Let c = +s)/+1, so the complete multipatite gaph we ae woking in is K c d = K +s+1). Let C = G 1 + G + 1, + s + 1) and C s = G 3 + G + + 6s + 1, 6 + 8s + 3) whee G 1 = Q0, + s +, 1), G = P 1, 1, ), G 3 = Q + s +, 6 + 6s +, s 1), G = P + 5s + 1, 6 + 5s + 1, s). Fist, we show that G 1 + G + 1, + s + 1) is a cycle of length and G 3 + G + + 6s + 1, 6 + 8s + 3) is a cycle of length s. Note that by Q1) and P1), the fist vetex of G 1 is + s + 1, and the last is 1; the fist vetex of G is 1, and the last is 1; the fist vetex of G 3 is 6 + 8s + 3, and the last is + 5s + 1; and the fist vetex of G is + 5s + 1, and the last is + 6s + 1. Fo 1 i, let A i and B i denote the sets labeled A and B in Q) and P) coesponding to the path G i. Then using Q) and P), we compute A 1 = [0, 1], B 1 = [3 + s +, + s + 1], A = [ 1, 1], B = [, 3 1], A 3 = [ + s +, + 5s + 1], B 3 = [6 + 7s +, 6 + 8s + 3], A = [ + 5s + 1, + 6s + 1], B = [6 + 6s +, 6 + 7s + 1]. Thus, A 1 A < B < B 1 < A 3 A < B < B 3. Note that V G 1 ) V G ) = { 1} and V G 3 ) V G ) = { + 5s + 1}; othewise, G i and 7

8 G j ae vetex-disjoint fo i j. Theefoe, G 1 + G + 1, + s + 1) is a cycle of length and G 3 + G + + 6s + 1, 6 + 8s + 3) is a cycle of length s. Next, let E i denote the set of edge labels in G i fo 1 i. By Q3) and P3), we have edge labels E 1 = [ + s + 3, + s + 1], E 3 = [ + s + 3, + s + 1], E = [1, ], E = [ + 1, + s]. Moeove, the path 1, + s + 1) consists of an edge with label + s +, and the path + 6s + 1, 6 + 8s + 3) consists of an edge with label + s+. Thus, the edge set of G has one edge of each label i whee 1 i + s + 1 except + s + 1. That is, the set of edge labels is [1, cd/ ] \ cz. Theefoe, we have a -modula ρ-labeling of G. Case : d = 8. Let c = +s)/8+1, so the complete multipatite gaph we ae woking in is K c d = K +s+1) 8. Without loss of geneality, we can assume that s. Case.1: + s is even. Let C = G 1 + G + 1, + s + 3) and C s = G 3 + G + G 5 + G s +, 6 + 8s + 7) whee G 1 = Q0, + s +, 1), G = P 1, 1, ), G 3 = Q + s +, 7 + 7s + 7, s ), G = Q +s s + 5, +s s + 8, + s 1), G 5 = P + 5s +, 5 + 6s + 5, + s), G 6 = P +s + + 5s +, +s s +, s ). If we continue as in the poof fo Case 1, we can see that we have an 8-modula ρ-labeling of G. Case.: + s is odd. Let C = G 1 + G + 1, + s + 3) and C s = G 3 + G + G 5 + G s +, 6 + 8s + 7) whee G 1 = Q0, + s +, 1), G = P 1, 1, ), G 3 = Q + s +, 7 + 7s + 7, s ), G = P +s s +, +s s + 7, + s 1), G 5 = P + 5s + 3, 5 + 6s +, + s), G 6 = Q +s s + 5, +s s + 5, s ). 8

9 If we continue as in the poof fo Case 1, we can see that we have an 8-modula ρ-labeling of G. Case 3: d = + s. Let c = + s)/ + s) + 1, so the complete multipatite gaph we ae woking in is K c d = K 9 +s). Case 3.1: s 0 mod ). Let C = G 1 + G + 9, s 1) and C s = G 3 + G s, 7 + 9s 1) whee G 1 = 1 Q5i 5, s i 5, 8)) + Q5 5, s, 7), G = P 5 + i 6, 7 5i 6, 8)), G 3 = s 1 Q9 + 9 s + 5i 5, 7 + 9s i 5, 8)) + Q9 + 3 s 5, 7 + 8s, 7), G = s P s + i 6, 7 + 8s 5i 6, 8)). Fist, we show that G 1 + G + 9, s 1) is a cycle of length and G 3 + G s, 7 + 9s 1) is a cycle of length s. Note that by Q1) and P1), the fist vetex of G 1 is 9 +9 s 1, and the last is 5 ; the fist vetex of G is 5, and the last is 9 ; the fist vetex of G 3 is 7 + 9s 1, and the last is s ; and the fist vetex of G is s, and the last is s. Fo 1 i, let A i and B i denote the sets labeled A and B in Q) and 9

10 P) coesponding to the path G i. Then using Q) and P), we compute A 1 = B 1 = 1 1 = [7 A = ] [0, 5 ], [5i 5, 5i ]) [5 5, 5 [9 + 9 s i 1, s i + 3]) [7 + 9 s, s + 3] + 9 s, s 1], B = A 3 = s 1 [9 B 3 = s [5 [7 + i 6, 5 + i ]) = [5, 9 ], 5i 1, 7 5i + ]) [9 1, 7 3], [9 + 9 s + 5i 5, s + 5i ]) [9 + 3 s 5, s ] + 9 s, s ], 1 [7 + 9s i 1, 7 + 9s i + 3]) + 8s, 7 + 8s + 3] [7 = [7 + 8s, 7 + 9s 1], A = s = [9 B = s [9 + 3 s + i 6, s + i ]) + 3 s, s ], [7 + 8s 5i 1, 7 + 8s 5i + ]) 1, 7 + 8s 3]. [7 + 7 s Thus, A 1 A < B < B 1 < A 3 A < B < B 3. Note that V G 1 ) V G ) = {5 } and V G 3) V G ) = {9 +3 s }; othewise, G i and G j ae vetex-disjoint fo i j. Theefoe, G 1 +G +9, 9 +9 s 1) is a cycle of length and G 3 + G s, 7 + 9s 1) is a cycle of length s. Next, let E i denote the set of edge labels in G i fo 1 i. By Q3) 10

11 and P3), we have edge labels E 1 = 1 [9 + 9 s 9i + 1, s 9i + 8]) [9 + 9 s +, s + 8] = [9 + 9 s +, s 1] \ {9 + 9 s + 9, s E = = [1, 9 E 3 = s 1 [9 9i + 1, 9 9i + 8]) 1] \ {9, 18,..., 9 9}, + 18,..., s 9}, [9 + 9 s 9i + 1, s 9i + 8]) [9 + 9 s +, s + 8] = [9 + 9 s +, s 1] \ {9 + 9 s + 9, s E = s + 18,..., s 9}, [9 + 9 s 9i + 1, s 9i + 8]) = [9 + 1, s 1] \ {9 + 9, ,..., s 9}. Moeove, the path 9, s 1) consists of an edge with label s + 1, and the path s, 7 + 9s 1) consists of the edge with label s + 1. Thus, the edge set of G has one edge of each label i, whee 1 i s 1 except 9, 18,..., s 9. That is, the set of edge labels is [1, cd/ ] \ cz. Theefoe, we have an + s)-modula ρ-labeling of G. Case 3.: 0 and s 1 mod ). If s = 1, let C s = 7 +9, 9 +5, 7 +7, 9 +6, 7 +9). Othewise, let C = G 1 + G + 9 1, s 1 + ) and C s = G s 1 s 1 +5, 7 +8s 1, )+G +9 s , 7 +9s) whee G 1 = Q0, s 1, ) + 1 Q5i, s 1 i, 8) ) + Q5, s 1 + 3, 3), G = P 5 + i 5, 7 5i 5, 8)), G 3 = Q9 + 9 s 1 + 5, 7 + 9s, ) + s 1 1 Q9 + 9 s 1 + 5i + 3, 7 + 9s i 6, 8)) G = s 1 + Q9 + 3 s 1 + 3, 7 + 8s, 5), P 9 s i +, 7 + 8s 5i 6, 8)). If we continue as in the poof fo Case 3.1, we can see that we have an + s)-modula ρ-labeling of G. 11

12 Case 3.3: 0 and s mod ). Let C = G 1 + G + 9, s + 8) and C s = G 3 + G + 9 s , 7 + 9s 1) whee G 1 = 1 Q5i 5, s i +, 8) ) + Q5 5, s + 5, 7), G = P 5 + i 6, 7 5i 6, 8)), G 3 = s Q9 + 9 s + 5i +, 7 + 9s i 5, 8)) s , 7 + 8s, 3), + Q9 G = P 9 + s s , 7 + 8s 6, ) P 9 s i + 8, 7 + 8s 5i 8, 8)). If we continue as in the poof fo Case 3.1, we can see that we have an + s)-modula ρ-labeling of G. Case 3.: 0 and s 3 mod ). Let C = G 1 + G + 9 1, ) and C s = G s + 1, 9 s ) + G + 9 s , 7 + 9s) whee G 1 = Q0, 9 + 9, ) + 1 Q5i, 9 i + 7, 8) ) + Q5, 7 + 1, 3), G = P 5 + i 5, 7 5i 5, 8)), G 3 = Q9 + 1, 7 + 9s, ) + s 3 G = P 9 + s 3 Q9 + 5i + 1, 7 + 9s i 6, 8)), s , 7 + 8s 7, 6) P 9 s i + 16, 7 + 8s 5i 8, 8)). If we continue as in the poof fo Case 3.1, we can see that we have an + s)-modula ρ-labeling of G. Case 3.5: s 1 mod ). If s = 1, let C s = , , , , ). Othewise, let C = G s 1 + 8, 5 1 1, 1 +, ) + G , s 1 + 8) and 1

13 C s = G 3 + G s , s + 6) whee G 1 = 1 G = 1 G 3 = Q9 1 Q5i 5, s 1 i +, 8) ), P i 3, 7 1 5i 3, 8) ), + 9 s 1 + 9, s, 6) + s 1 1 Q s 1 + 5i + 8, s i 1, 8) ) + Q s 1 + 8, s + 5, 3), G = P s 1 + 9, s + 1, ) + s 1 1 P s 1 + i + 7, s 5i 1, 8) ) + P s 1 + 7, s 1 + 9, 6). If we continue as in the poof fo Case 3.1, we can see that we have an + s)-modula ρ-labeling of G. Case 3.6: 1 and s mod ). If = 1, let C = 9 s + 13, 0,, 1, 9 s + 13). If s =, let C s = , , 7 1 +, , 7 1 +, , , , ). Othewise, let C = G , , 5 + 1) + G , s + 13) and C s = G s + 8, s + 16) + G s , 7 + 9s + 7, s 1 + 1, 7 + 9s + 6) whee G 1 = Q0, s + 9, ) Q5i, s i + 7, 8) ) + Q5 1, s + 10, 5), G = 1 P i 3, 7 1 5i 3, 8) ), G 3 = s Q s + 5i + 11, s i +, 8) ), G = P s + 16, s, 6) + s 1 P s + i + 15, s 5i 1, 8) ) + P s + 15, s + 17, 6). If we continue as in the poof fo Case 3.1, we can see that we have an + s)-modula ρ-labeling of G. Case 3.7: 1 and s 3 mod ). If s = 3, let C s = , , , , , 9 + 0, 7 + 8, 9 + 1, 7 + 7, 9 +, , 9 + 3, ). Othewise, let C = G

14 s , 5, 7 +, 5 +1)+G s 3 +1, ) and C s = G 3 + G s 3 + 3, s + 6) whee G 1 = 1 G = 1 G 3 = Q9 1 G = s 3 Q5i 5, 9 1 i + 13, 8) ), P i 3, 7 1 5i 3, 8) ), + 18, s, 6) + s 3 1 Q i + 17, s i 1, 8) ) + Q s , s + 3, 7), P s 3 + i + 16, s 5i + 1, 8) ) + P s 3 + 0, s 3 +, 6). If we continue as in the poof fo Case 3.1, we can see that we have an + s)-modula ρ-labeling of G. Case 3.8: s mod ). If s =, let C s = , , , , 7 + 7, 9 + 0, 7 + 6, 9 + 1, ). Othewise, let C = G 1 + G , s + 17) and C s = G 3 + G s + 1, 7 + 9s + 13) whee G 1 = Q5i 5, s i + 13, 8) ) + Q5, s + 1, 3), G = P 5 + G 3 = Q9 G = s + 1, 7 + 1, ) P 5 + i 1, 7 5i 1, 8) ), + 9 s + 18, 7 + 9s + 9, ) + s 1 Q9 + 9 s + 5i + 16, 7 + Q9 + 3 s + 16, 7 + 8s + 10, 7), + 9s i + 7, 8) ) P s + i + 15, 7 + 8s 5i + 8, 8) ) + P s + 15, , ). If we continue as in the poof fo Case 3.1, we can see that we have an + s)-modula ρ-labeling of G. Case 3.9: and s 3 mod ). If =, let C = 9 s 3 +, 0, 9 s 3 + 1, 1, 5,,, 3, 9 s 3 + ). Othewise, let C = G 1 + G , 9 + ) and C s = G s 3 + 5, 7 + 8s + 1, s 3 + 1

15 6) + G s 3 + 8, 7 + 9s + 13) whee G 1 = Q0, , ) + 1 Q5i, 9 i + 16, 8) ) + Q5, , 7), G = P 5 + 1, 7 + 1, ) + P 5 + i 1, 7 5i 1, 8) ), G 3 = s 3 Q9 + 5i + 18, 7 + 9s i + 9, 8) ) + Q9 + 3 s 3 + 3, 7 + 8s + 11, 5), G = s 3 P s 3 + i +, 7 + 8s 5i + 7, 8) ) + P s 3 + 6, s , ). If we continue as in the poof fo Case 3.1, we can see that we have an + s)-modula ρ-labeling of G. Case 3.10: s 3 mod ). Let C = G 1 + G , ) and C s = G 3 + G s 3 + 3, s 3 + 0, s , 7 3 9s + 0, , s + 19) whee G 1 = 3 Q5i 5, 9 3 i +, 8) ) + Q5 3, , 5), G = P 5 3 +, 7 3 +, 6) G 3 = s 3 P i + 1, 7 3 5i + 1, 8) ), Q i +, s i + 15, 8) ) Q s 3 + 9, s + 19, 3), G = P s , s + 15, ) + s 3 P s 3 + i + 8, s 5i + 13, 8) ). If we continue as in the poof fo Case 3.1, we can see that we have an + s)-modula ρ-labeling of G. Case : d = + s). Let c = + s)/ + s) + 1, so the complete multipatite gaph we ae woking in is K c d = K 5 +s). Case.1: is odd, s is odd. If s = 1, let C s = , 5 + 5, , 5 + 6, ). 15

16 Othewise, let C = G s, 3 1 1,, 3 + 1) + G , 5 + 5s 1) and C s = G s 1 + 9s + 9, , 15 1 s 1 + 9s + 8, ) + G s 1 1 s 1 + 5, , s 1 + 6, s + 7, 5 + 5s, s + 6) whee G 1 = 1 Q3i 3, 5 + 5s i 3, ), G = 1 1 P 3 + i 1, 3i 5, ), G 3 = s 1 1 Q5 + 5s + 3i 1, s i +, ), G = s 1 s 1 P i + 3, s 3i +, ). If we continue as in the poof fo Case 3.1, we can see that we have a + s)-modula ρ-labeling of G. Case.: is odd, s is even. Let C = G 1 ++5s, 3 1 1,, 3 +1)+G , 5+5s 1) and C s = G s + 8, s 1 1, s + 6, s ) + G s 1, s s 1 + 7, , s + 7, 5 + 5s, s + 6) whee G 1 = 1 Q3i 3, 5 + 5s i 3, ), G = 1 G 3 = s 1 G = s 1 P i 1, 3i 5, ), Q5 + 5s + 3i 1, s i +, ), s 1 P i, s 3i + 3, ). If we continue as in the poof fo Case 3.1, we can see that we have a + s)-modula ρ-labeling of G. Case.3: is even, s is odd. Let C = G 1 ++5s+1, 3 3, +5s, 3 )+G +5, 5+5s 1) and C s = G s 1 s 1 + 9s, , s, ) + G s 1 + 6, s 1) whee G 1 = 1 Q3i 3, 5 + 5s i 3, ), G = P 3 + i, 3i, ), G 3 = s 1 Q5 + 5s + 3i 3, s i 3, ), G = s 1 s 1 P i +, s 3i 5, ). If we continue as in the poof fo Case 3.1, we can see that we have a + s)-modula ρ-labeling of G. 16

17 Case.: is even, s is even. Let C = G 1 ++5s+1, 3 3, +5s, 3 )+G +5, 5+5s 1) and C s = G s + 9s + 1, , s, ) + G s, s 1) whee G 1 = 1 Q3i 3, 5 + 5s i 3, ), G = P 3 + i, 3i, ), G 3 = s 1 Q5 + 5s + 3i 3, s i 3, ), G = s s P i, s 3i, ). If we continue as in the poof fo Case 3.1, we can see that we have a + s)-modula ρ-labeling of G. Case 5: d = + s). Let c = + s)/ + s) + 1, so the complete multipatite gaph we ae woking in is K c d = K 3 +s). Let C = G s, ) + G + 3, 6 + 6s 1) and C s = G s 1, 6 + 8s + 1) + G s, 9 + 1s 1) whee G 1 = 1 Qi, 6 + 6s i, ), G = P 3 + i, 5 3 i, ), G 3 = s Q6 + 6s + i, 9 + 1s i, ), G = s 1 P 6 + 8s + i, s i 3, ). In the case when = 1, the path G 1 is empty, and when s = 1, the path G is empty. Howeve, this does not change the poof in any way.) If we continue as in the poof fo Case 3.1, we can see that we have a + s)-modula ρ-labeling of G. Theoem 9. Let G = C C s and let n = + s. Then thee exists a cyclic G-decomposition of K n+1) t, K n+1) t, K n/+1) t, K n/+1) 8t, K 9 n/)t, K 5 n/)t, K 3 nt, and of K nt fo evey positive intege t. Lemma 10. A d-modula ρ-labeling of C C s+ exists fo, s 1 and d {1,, + s + 1, + s + 1)}. Poof. Let G = C C s+ whee, s 1. The cases d = 1 and d = + s + 1) can be obtained fom the fact that such a G necessaily admits a ρ + -labeling see []). Case 1: d =. Let c = + s + )/ + 1, so that the complete multipatite gaph we ae woking in is K c d = K +s+). 17

18 Case 1.1: s. Let C = G 1 + G + 1, ) and C s+ = G 3 + G + G s +, 8 + s + ) whee G 1 = Q0, + 1, 1), G = P 1, 1, ), G 3 = Q + 1, 8 + s + 3, s + 1), G = P + s + 1, 6 + 3s + 3, 1), G 5 = Q5 + s +, 9 + s +, s + 1). If we continue as in the poof fo Case 1 in Lemma 1, we can see that we have a -modula ρ-labeling of G. Case 1.: > s. Let C = G 1 + G + G 3 + 1, +, 0) and C s+ = G + G s + 5, + s +, 8 + s + 6) whee G 1 = P 0, + s +, s ), G = P s 1, 3 s +, s ), G 3 = P,, + ), G = Q + 3, 8 + s + 5, s + 1), G 5 = P + s + 3, 8 + s + 5, s 1). If we continue as in the poof fo Case 1 in Lemma 1, we can see that we have a -modula ρ-labeling of G. Case : d = + s + 1. Let c = + s + )/ + s + 1) + 1, so the complete multipatite gaph we ae woking in is K c d = K 5 +s+1). In ode to show that G admits a d-modula ρ-labeling, we examine when is odd o even and when s is odd o even and show that any of the fou possible combinations will satisfy the necessay conditions fo the desied labeling. Case.1: is odd. Let C = G s +, s + 9, 9 + 5s +, s + 10) + G s + 10, s + 3) whee G 1 = 1 Q5 + 5s + 3i + 1, s i + 1, ), G = 1 1 P s + i + 8, 9 + 5s 3i 1, ). If we continue as in Case 3.1 in Lemma 1, we can see that the set of edge labels is [1, 5 1] \ cz with 5 + 5s + V C ) s

19 Case.: is even. Let C = G s + 5, s + 1, 9 + 5s +, s + ) + G s +, s + 3) whee G 1 = 1 Q5 + 5s + 3i + 1, s i + 1, ), G = P s + i, 9 + 5s 3i, ). If we continue as in Case 3.1 in Lemma 1, we can see that the set of edge labels is [1, 5 1] \ cz with 5 + 5s + V C ) s + 3. Case.3: s is odd. Let C s+ = G s +, 3 s 1 + 3, 5 + s + 1, 3 s 1 + ) + G + 5 s 1 +, 5 + 5s + 3, 0, 5 + 5s + 1) whee G 3 = s 1 Q3i 1, 5 + 5s i 1, ), G = s 1 s 1 P 3 + i +, 5 + s 3i, ). If we continue as in Case 3.1 in Lemma 1, we can see that the set of edge labels is [5 + 1, cd 1)/ ] \ cz with 0 V C s ) 5 + 5s + 3. Case.: s is even. Let C s+ = G s + 3, 3 s 5s + 3, 0, 5 + 5s + 1) whee G 3 = s 1 1, 5 + s + 1, 3 s ) + G + 5 s, 5 + Q3i 1, 5 + 5s i 1, ), G = s P 3 s + i, 5 + s 3i, ). If we continue as in Case 3.1 in Lemma 1, we can see that the set of edge labels is [5 + 1, cd 1)/ ] \ cz with 0 V C s ) 5 + 5s + 3. Since a labeling of C fom eithe of the fist two subcases will be vetex disjoint fom a labeling of C s+ fom eithe of the last two subcases, we have a labeling of G = C C s+ whee the set of edge labels is [1, cd/ ] \ cz. Theefoe, we have a + s + 1)-modula ρ-labeling of G. Theoem 11. Let G = C C s+ whee and s ae positive integes and and let n = + s +. Then thee exists a cyclic G-decomposition of K n+1) t, K n/+1) t, K 5 n/)t, and of K nt fo evey positive intege t. Befoe poceeding to ou final case, we note that the paity condition i.e., Lemma 5) ules out the existence of a d-modula ρ-labelings of G in Lemma 10 fo d = and fo d = + s +. 19

20 Lemma 1. A d-modula ρ-labeling of C + C s+ exists fo, s 1 and d {1,,, 8, + s + 1, + s + 1), + s + 1), 8 + s + 1)}. Poof. Let G = C + C s+ whee 1 s. The cases d = 1, d =, and d = 8 + s) can be obtained fom the fact that such a G necessaily admits an α-labeling see [1]). Case 1: d =. Let c = + s + )/ + 1, so that the complete multipatite gaph we ae woking in is K c d = K +s+3). Case 1.1: = s. If = s = 1, let C + = 0, 3,, 6,, 9, 0) and C s+ = 10,, 11, 19, 13, 3, 10). We leave it to the eade to check that this yields a -modula ρ-labeling of G. If = s > 1, let C + = G 1 + G + + 1, + 5, 0) and C s+ = G 3 + G + 6s + 5, 10s + 9, 6s + 7, 1s + 11) whee G 1 = P 0, +, 3), G = Q,, + 3), G 3 = Qs + 6, 10s + 10, s + 1), G = P 5s + 6, 9s + 11, s ). If we continue as in the poof fo Case 1 in Lemma 1, we can see that we have a -modula ρ-labeling of G. Case 1.: < s. Let C + = G 1 + G + + 1, + 3, 0) and C s+ = G 3 + G + G s + 7, + s + 5, 8 + s + 9) whee G 1 = P 0, +, 1), G = Q + 1, + 1, + 1), G 3 = Q +, 8 + s + 8, s + 1), G = P + s +, 6 + 3s + 7, ), G 5 = P 5 + s +, 9 + s + 7, s 1). If we continue as in the poof fo Case 1 in Lemma 1, we can see that we have a -modula ρ-labeling of G. Case : d = 8. Let c = + s + )/8 + 1, so that the complete multipatite gaph we ae woking in is K c d = K +s+) 8. Case.1: = s. Let C + = G 1 + G , +, 8 + 7) and C s+ = G , ) + G , , 8 + 8) whee G 1 = Q0, 6 + 6, + 1), G = P, 5 + 5, 1), G 3 = P 8 + 8, , ), G = Q9 + 9, 9 + 9, + 1). 0

21 If we continue as in the poof fo Case 1 in Lemma 1, we can see that we have an 8-modula ρ-labeling of G. Case.: < s < and + s is odd. Let C + = G 1 + G + G s + 5, + 1, + s + 7) and C s+ = G + G 5 + G 6 + G s + 9, + 8s + 13, + s + 8) whee G 1 = Q0, + s + 6, + 1), G = P, + 3s + 6, s 1), G 3 = P +s 1, +s 1 + s + 5, 3 s), G = P + s + 8, 6 + 6s + 1, s 1), G 5 = Q3 + 5s + 9, 3 + 7s + 13, 1), G 6 = P + 5s + 8, 5 + 6s + 10, s + 1), G 7 = P 7 +s 1 + s + 1, 7 +s 1 + s + 1, + s + 1). If we continue as in the poof fo Case 1 in Lemma 1, we can see that we have an 8-modula ρ-labeling of G. Case.3: < s < and + s is even. Let C + = G 1 + G + G , + s + 7) and C s+ = G + G 5 + G 6 + G s + 10, + 8s + 1) whee G 1 = Q0, + s + 6, + 1), G = P, + 3s + 6, s 1), G 3 = Q +s + 1, +s + s + 5, 3 s + 1), G = Q + s + 8, 6 + 6s + 1, s ), G 5 = Q3 + 5s + 9, 3 + 7s + 11, + 1), G 6 = P + 5s + 9, 5 + 6s + 11, s 1), G 7 = Q7 +s + s + 10, 7 +s + s + 10, + s + 1). If we continue as in the poof fo Case 1 in Lemma 1, we can see that we have an 8-modula ρ-labeling of G. Case.: s = Let C + = G 1 + G + 1, 1 + 9, + 1, ) and C s+ = G 3 + G + G 5 + G , 8 + 3, ) whee G 1 = Q0, , + 1), G = P, , ), G 3 = P , + 18, + 1), G = Q18 + 1, + 1, ), G 5 = Q19 + 1, , + 3), G 6 = P , , + ). If we continue as in the poof fo Case 1 in Lemma 1, we can see that we have an 8-modula ρ-labeling of G. 1

22 Case.5: s > and + s is odd. Let C + = G 1 + G + + s + 6, + 1, + s + 7) and C s+ = G 3 + G + G 5 + G 6 + G s + 10, + 8s + 1, + s + 8) whee G 1 = Q0, + s + 6, + 1), G = P, + s + 6, 1), G 3 = P + s + 8, 7 + 7s + 1, s 3 ), G = Q5 +s 1 + s + 11, p +s 1 + s + 17, + s + 1), G 5 = Q3 + 5s + 10, 3 + 7s + 1, 1), G 6 = P + 5s + 9, 5 + 6s + 11, s + 1), G 7 = P 7 +s 1 + s + 13, 7 +s 1 + s + 13, + s + 1). If we continue as in the poof fo Case 1 in Lemma 1, we can see that we have an 8-modula ρ-labeling of G. Case.6: s > and + s is even. Let C + = G 1 + G + + s + 6, + 1, + s + 7) and C s+ = G 3 + G + G 5 + G 6 + G s + 10, + 8s + 1, + s + 8) whee G 1 = Q0, + s + 6, + 1), G = P, + s + 6, 1), G 3 = P + s + 8, 7 + 7s + 1, s 3 ), G = P 5 +s + s + 7, 9 +s + s + 11, + s + 1), G 5 = Q3 + 5s + 9, 3 + 7s + 13, 1), G 6 = P + 5s + 8, 5 + 6s + 10, s + 1), G 7 = Q7 +s + s + 10, 7 +s + s + 10, + s + 1). If we continue as in the poof fo Case 1 in Lemma 1, we can see that we have an 8-modula ρ-labeling of G. Case 3: d = + s + 1. Let c = + s + )/ + s + ) + 1, so that the complete multipatite gaph we ae woking in is K c d = K 9 +s+1). Case 3.1: and s ae both odd. In ode to show that G admits a d-modula ρ-labeling, we examine when 1, 3 mod ) and when s 1, 3 mod ) and show that any of the fou possible combinations will satisfy the necessay conditions fo the desied labeling. Case 3.1.1: 1 mod ). If = 1, let C + = 0, 9 s 1 s 1 s 1 +1, 1, 9 +9, 3, 9 +13, 0). We leave it to the eade to check that this yields an + s + 1)-modula ρ-labeling of G.

23 If > 1, let C + = G 1 + G , 9 +s 9 1, 9 1 3, 9 +s + ) whee G 1 = Q0, 9 +s 1, ) Q5 1, 7 +s + s, 7), G = 1 Case 3.1.: 3 mod ). Let C + = G 1 +G +9 3 whee G 1 = Q0, 9 +s Q5i, 9 +s i, 8) ) P i 3, 7 +s + s 5i, 8) ). +6, 9 +s s, 9 +8, 9 +) 3, ) + Q5i, 9 +s i, 8) ) + Q , 7 +s + s +, 3), G = P 5 3 +, 7 +s + s, ) + 3 P i +, 7 +s + s 5i, 8) ). Case 3.1.3: s 1 mod ). If s = 1, let C s+ = , , , , , , ). We leave it to the eade to check that this yields an + s + 1)-modula ρ-labeling of G. If s > 1, let C s+ = G 3 + G + 9 +s + 9 s 1 + 8, 9 +s + 9 s 1 + 1, 9 +s + 5) whee G 3 = P 9 +s + 5, 9 +s + 9 s 1 + 5, 5) + s 1 1 Q9 +s + 5i +, 9 +s + 9 s 1 i +, 8) ) + Q9 +s + 5 s 1 +, 9 +s + 7 s 1 + 8, ), G = Q9 +s + 5 s 1 + 7, 9 +s + 7 s 1 + 7, 3) + s 1 P 9 +s + 5 s 1 + i +, 9 +s + 7 s 1 5i +, 8) ). Case 3.1.: s 3 mod ). Let C s+ = G 3 +G +9 +s whee +9 s 3 +13, 9 +s G 3 = P 9 +s + 5, 9 +s + 1, 5) + s 3 G = Q9 +s + s 3 s 3 +s +9 +1, 9 +5) Q9 +s + 5i +, 9 +s i + 13, 8) ), + 10, 9 +s + 7 s , 7) + 5 s 3 P 9 +s + 5 s 3 + i + 9, 9 +s + 7 s 3 5i + 9, 8) ). 3

24 If we continue as in the poof fo Case in Lemma, we can see that we have an + s + 1)-modula ρ-labeling of G. Case 3.: and s ae both even. In ode to show that G admits a d-modula ρ-labeling, we examine when 0, mod ) and when s 0, mod ) and show that any of the fou possible combinations will satisfy the necessay conditions fo the desied labeling. Case 3..1: 0 mod ). Let C + = G 1 + G + 9 +s 9 + 3, 9 +s + 1, 9 + ) whee G 1 = Q0, 9 +s, ) + 1 Q5i, 9 +s i, 8) ) + Q5 G = P 5, 7 +s P 9 +s, 7 + s + 1, 5), + s, ) P 5 +s + i 3, 7 3, 9 +s 9, 5). + s 5i 3, 8) ) Case 3..: mod ). If =, let C + = 0, 9 s s s s s +1, 1, 9 +11, 3, 9 +9,, 9 +8, 6, 9 +13, 0). We leave it to the eade to check that this yields an + s + 1)-modula ρ-labeling of G. If >, let C + = G s + s +, 5 + 3) + G + 9 +s 9 1, 9 + 6, 9 +s + ) whee G 1 = Q0, 9 +s, ) + Q5i, 9 +s i, 8) ), G = P 5 + 3, 7 +s + s, 6) + 1 P 5 + i +, 7 +s + s 5i 5, 8) ) + P 9 +, 9 +s 9, 5). Case 3..3: s 0 mod ). Let C s+ = G s + 7 s +s + 7, s + 6) + G + 9 +s 9 s +s + 6, s +s + 8, 9 + 5, 9 +s + 9 s + 6) whee G 3 = s 1 G = s Q9 +s + 5i +, 9 +s + 9 s i +, 8)) + 5 s +s +, s +, 6), + Q9 +s Case 3..: s mod ). Let C s+ = G s P 9 +s + 5 s +s + i +, s 5i +, 8)). +7 s +15, 9 +s +5 s + +s s +7, )+

25 G +9 +s whee +9 s +11, 9 +s +9 s +17, 9 +s +s s +5, ) G 3 = s Q9 +s + 5i +, 9 +s + 9 s i + 11, 8) ), G = Q9 +s + 5 s + 9, 9 +s + 7 s + 9, 5) + s P 9 +s + 5 s + i + 7, 9 +s + 7 s 5i + 7, 8) ). If we continue as in the poof fo Case in Lemma, we can see that we have an + s + 1)-modula ρ-labeling of G. Case 3.3: + s is odd. Fo this case, we elax the condition that s. Then without loss of geneality, we need only conside when is odd and s is even. In ode to show that G admits a d-modula ρ-labeling, we examine when 1, 3 mod ) and when s 0, mod ) and show that any of the fou possible combinations will satisfy the necessay conditions fo the desied labeling. Case 3.3.1: 1 mod ). If = 1, let C + = 0, 9 s s s + 7, 1, 9 + 5, 3, 9 + 8, 0). We leave it to the eade to check that this yields an + s + 1)-modula ρ-labeling of G. If > 1, let C + = G 1 +G +9 +s s 1 +5, 9 +3, 9 +8) whee G 1 = 1 Q5i 5, 9 +s 1 i +, 8) ) + Q5 1, 7 +s 1 + s + 5, 3), G = P , 7 +s 1 + s, ) P i 1, 7 +s 1 + s 5i 1, 8) ) + P 9 1 1, 9 +s , 5). Case 3.3.: 3 mod ). Let C + = G 1 + G + 9 +s 1 9 3, , 9 +s 1 + 8) whee G 1 = 3 Q5i 5, 9 +s 1 i +, 8) ) + Q5 3, 7 +s 1 + s + 3, 7), G = 3 P i 1, 7 +s 1 + s 5i + 1, 8) ) + P , 9 +s , 5). Case 3.3.3: s 0 mod ). Let C s+ = G s s +s , s + 10) + G + 9 5

26 +s s whee G 3 = s 1 G = s +s , s +s 1 + 1, 9 + 9, 9 +s s + 10) Q9 +s 1 + 5i + 6, 9 +s s i + 6, 8)) + 5 s +s 1 + 6, s + 8, 6), + Q9 +s 1 P 9 +s s +s 1 + i + 6, s 5i + 6, 8)). Case 3.3.: s mod ). Let C s+ = G s 1 +7 s +s 1 +19, 9 +5 s +s 1 +11, s + 18) + G + 9 +s s + 15, 9 +s s + 1, 9 +s 1 + 9, 9 +s s + 19) whee G 3 = s Q9 +s 1 + 5i + 6, 9 +s s i + 15, 8) ), G = Q9 +s s + 13, 9 +s s + s + 13, 5) P 9 +s s + i + 11, 9 +s s 5i + 11, 8) ). If we continue as in the poof fo Case in Lemma, we can see that we have an + s + 1)-modula ρ-labeling of G. Case : d = + s + 1). Let c = + s + )/ + s + ) + 1, so that the complete multipatite gaph we ae woking in is K c d = K 5 +s+1). In ode to show that G admits a d-modula ρ-labeling, we examine when is even o odd and when s is even o odd and show that any of the fou possible combinations will satisfy the necessay conditions fo the desied labeling. Case.1: is odd. Let C + = G 1 + G , s + 5, , 5 + 5s + ) whee G 1 = 1 G = 1 1 Q3i 3, 5 + 5s i +, )) + Q3, + 5s +, 3), P i 1, + 5s 3i, ) ). Case.: is even. Let C + = G s +, 3, + 5s +, 3 + 1) + G + 5 1, 5 + 5s + 3, 5 + 1, 5 + 5s + ) whee G 1 = Q3i 3, 5 + 5s i +, )), G = 1 P 3 + i 1, + 5s 3i 1, )). Case.3: s is odd. Let C s+ = G 3 +G +15 s , 5+10s+8, 5+5s+5, 5+10s+6) 6

27 whee G 3 = s 1 Q5 + 5s + 3i +, s i +, )), G = Q13 s+1 + 5, 5 + 9s +, 3) + s 1 P 13 s i 1, 5 + 9s 3i + 3, ) ). Case.: s is even. Let C s+ = G s + 8, 13 s s + 5 +, 5 + 9s + 7, ) + G + 15 s , s + 8, 5 + 5s + 5, s + 6) whee G 3 = s 1 G = s Q5 + 5s + 3i +, s i +, )), P 13 s i +, 5 + 9s 3i +, )). If we continue as in the poof fo Case in Lemma, we can see that we have a + s + )-modula ρ-labeling of G. Case 5: d = + s + 1). Let c = + s + )/ + s + ) + 1, so that the complete multipatite gaph we ae woking in is K c d = K 3 +s+). If s = 1, let C + = 0, 16,, 1,, 17, 0) and C s+ = 18, 0, 19, 6,, 9, 18). We leave it to the eade to check that this yields a + s + )-modula ρ-labeling of G. If s > 1, let C + = G s + 5, ) + G + 3 1, 3 + 6s + 3, 3 + 1, 6 + 6s + 5) and C s+ = G s + 9, 6 + 8s + 5) + G s + 6, 6 + 1s + 11, 6 + 6s + 6, 6 + 1s + 7) whee G 1 = Qi, 6 + 6s i +, ), G = 1 P + i 1, 5 + 6s i +, ), G 3 = s Q6 + 6s + i + 6, 6 + 1s i + 6, ), G = s+1 P 6 + 8s + i +, s i + 7, ). If we continue as in the poof fo Case 3.1 in Lemma 1, we can see that we have a + s + )-modula ρ-labeling of G. Theoem 13. Let G = C + C s+ whee and s ae positive integes and let n = + s +. Then thee exists a cyclic G-decomposition of K n+1) t, K n+1) t, K n/+1) t, K n/+1) 8t, K 9 n/)t, K 5 n/)t, K 3 nt, and of K nt fo evey positive intege t. 5 Acknowledgement and Final Note This wok was done unde the supevision of the thid and fouth authos as pat of REU Site: Mathematics Reseach Expeience fo Pe-sevice and 7

28 fo In-sevice Teaches at Illinois State Univesity. The affiliations of the emaining authos at the time wee as follows: A. Su: Univesity High School Nomal, IL); J. Buchanan and E. Spaks: Illinois State Univesity; E. Pelttai: Nothen Illinois Univesity; G. Rasmuson: Roanoke-Benson High School Roanoke, IL); S. Tagais: Illinois Wesleyan Univesity. Refeences [1] J. Abham and A. Kotzig, Gaceful valuations of -egula gaphs with two components, Discete Math ), [] P. Adams, D. Byant, and M. Buchanan, A suvey on the existence of G-designs, J. Combin. Des ), [3] A. Benini and A. Pasotti, Decompositions of complete multipatite gaphs via genealized gaceful labelings, pepint. [] A. Blinco and S. I. El-Zanati, A note on the cyclic decomposition of complete gaphs into bipatite gaphs, Bull. Inst. Combin. Appl. 0 00), [5] D. Byant and S. El-Zanati, Gaph decompositions, in Handbook of Combinatoial Designs, C. J. Colboun and J. H. Dinitz Editos), nd ed., Chapman & Hall/CRC, Boca Raton, 007, pp [6] M. Buatti, Recusive constuctions fo diffeence matices and elative diffeence families, J. Combin. Des ), [7] M. Buatti and L. Gionfiddo, Stong diffeence families ove abitay gaphs, J. Combin. Des ), [8] M. Buatti and A. Pasotti, Gaph decompositions with the use of diffeence matices, Bull. Inst. Combin. Appl ), 3 3. [9] S. I. El-Zanati, C. Vanden Eynden, and N. Punnim, On the cyclic decomposition of complete gaphs into bipatite gaphs, Austalas. J. Combin. 001), [10] S. I. El-Zanati and C. Vanden Eynden, On Rosa-type labelings and cyclic gaph decompositions, Mathematica Slovaca ), [11] J. A. Gallian, A dynamic suvey of gaph labeling, Electon. J. Combin. 01), #DS6. [1] G. Ge, Goup divisible designs, in Handbook of Combinatoial Designs, C. J. Colboun and J. H. Dinitz Editos), nd ed., Chapman & Hall/CRC, Boca Raton, 007, pp

29 [13] A. Pasotti, On d-gaceful labelings, As Combin ), [1] A. Rosa, On cetain valuations of the vetices of a gaph, in Theoy of Gaphs Intenat. Sympos., Rome, 1966), ed. P. Rosenstiehl, Dunod, Pais; Godon and Beach, New Yok, 1967, pp [15] D. Sotteau, Decomposition of K m,n K m,n) into cycles cicuits) of length k, J. Combin. Theoy, Se. B, ),