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

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 61790-50 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. A1063038

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

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

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.

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 6. 0 1 11 0 1 11 0 1 11 0 1 11 = 7 1 8 7 1 8 16 3 37 31 38 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

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/ + 1 + 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

3 5 5 6 13 1 11 a) P 3, 8, 5) 13 1 11 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

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 6 + + 6s +, 6 + 8s + 7) whee G 1 = Q0, + s +, 1), G = P 1, 1, ), G 3 = Q + s +, 7 + 7s + 7, s ), G = Q +s + 3 + s + 5, +s + 5 + 6s + 8, + s 1), G 5 = P + 5s +, 5 + 6s + 5, + s), G 6 = P +s + + 5s +, +s + 6 + 5s +, 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 6 + + 6s +, 6 + 8s + 7) whee G 1 = Q0, + s +, 1), G = P 1, 1, ), G 3 = Q + s +, 7 + 7s + 7, s ), G = P +s 1 + 3 + s +, +s 1 + 5 + 6s + 7, + s 1), G 5 = P + 5s + 3, 5 + 6s +, + s), G 6 = Q +s 1 + + 5s + 5, +s 1 + 6 + 5s + 5, s ). 8

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, 9 + 9 s 1) and C s = G 3 + G + 9 + 7 s, 7 + 9s 1) whee G 1 = 1 Q5i 5, 9 + 9 s i 5, 8)) + Q5 5, 7 + 9 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 9 + 3 s + i 6, 7 + 8s 5i 6, 8)). Fist, we show that G 1 + G + 9, 9 + 9 s 1) is a cycle of length and G 3 + G + 9 + 7 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 9 + 3 s ; and the fist vetex of G is 9 + 3 s, and the last is 9 + 7 s. Fo 1 i, let A i and B i denote the sets labeled A and B in Q) and 9

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, 9 + 9 s i + 3]) [7 + 9 s, 7 + 9 s + 3] + 9 s, 9 + 9 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, 9 + 9 s + 5i ]) [9 + 3 s 5, 9 + 3 s ] + 9 s, 9 + 3 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, 9 + 3 s + i ]) + 3 s, 9 + 7 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 + 9 + 7 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

and P3), we have edge labels E 1 = 1 [9 + 9 s 9i + 1, 9 + 9 s 9i + 8]) [9 + 9 s +, 9 + 9 s + 8] = [9 + 9 s +, 9 + 9 s 1] \ {9 + 9 s + 9, 9 + 9 s E = = [1, 9 E 3 = s 1 [9 9i + 1, 9 9i + 8]) 1] \ {9, 18,..., 9 9}, + 18,..., 9 + 9 s 9}, [9 + 9 s 9i + 1, 9 + 9 s 9i + 8]) [9 + 9 s +, 9 + 9 s + 8] = [9 + 9 s +, 9 + 9 s 1] \ {9 + 9 s + 9, 9 + 9 s E = s + 18,..., 9 + 9 s 9}, [9 + 9 s 9i + 1, 9 + 9 s 9i + 8]) = [9 + 1, 9 + 9 s 1] \ {9 + 9, 9 + 18,..., 9 + 9 s 9}. Moeove, the path 9, 9 + 9 s 1) consists of an edge with label 9 + 9 s + 1, and the path 9 + 7 s, 7 + 9s 1) consists of the edge with label 9 + 9 s + 1. Thus, the edge set of G has one edge of each label i, whee 1 i 9 + 9 s 1 except 9, 18,..., 9 + 9 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, 9 + 9 s 1 + ) and C s = G 3 + 9 + 3 s 1 s 1 +5, 7 +8s 1, 9 +3 +6)+G +9 s 1 +7 +6, 7 +9s) whee G 1 = Q0, 9 + 9 s 1, ) + 1 Q5i, 9 + 9 s 1 i, 8) ) + Q5, 7 + 9 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 1 + 3 + 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

Case 3.3: 0 and s mod ). Let C = G 1 + G + 9, 9 + 9 s + 8) and C s = G 3 + G + 9 s + 7 + 1, 7 + 9s 1) whee G 1 = 1 Q5i 5, 9 + 9 s i +, 8) ) + Q5 5, 7 + 9 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 + 3 + 9, 7 + 8s, 3), + Q9 G = P 9 + s s + 3 + 10, 7 + 8s 6, ) P 9 s + 3 + 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, 9 + 13) and C s = G 3 + 7 + 8s + 1, 9 s 3 + 3 + 17) + G + 9 s 3 + 7 + 0, 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 3 + 3 + 17, 7 + 8s 7, 6) P 9 s 3 + 3 + 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 = 7 1 + 15, 9 1 + 9, 7 1 + 13, 9 1 + 10, 7 1 + 15). Othewise, let C = G 1 + 7 1 + 9 s 1 + 8, 5 1 1, 1 +, 5 1 + 1) + G + 9 1 + 1, 9 1 + 9 s 1 + 8) and 1

C s = G 3 + G + 9 1 + 7 s 1 + 10, 7 1 + 9s + 6) whee G 1 = 1 G = 1 G 3 = Q9 1 Q5i 5, 9 1 + 9 s 1 i +, 8) ), P 5 1 + i 3, 7 1 5i 3, 8) ), + 9 s 1 + 9, 7 1 + 9s, 6) + s 1 1 Q9 1 + 9 s 1 + 5i + 8, 7 1 + 9s i 1, 8) ) + Q9 1 + 3 s 1 + 8, 7 1 + 8s + 5, 3), G = P 9 1 + 3 s 1 + 9, 7 1 + 8s + 1, ) + s 1 1 P 9 1 + 3 s 1 + i + 7, 7 1 + 8s 5i 1, 8) ) + P 9 1 + 7 s 1 + 7, 7 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 + 5, 9 1 + 1, 7 1 +, 9 1 + 16, 7 1 +, 9 1 + 17, 7 1 + 1, 9 1 + 18, 7 1 + 5). Othewise, let C = G 1 + 5 1, 7 1 1 +, 5 + 1) + G + 9 1 1 + 1, 9 + 9 s + 13) and C s = G 3 + 7 1 + 8s + 8, 9 1 + 3 s + 16) + G + 9 1 + 7 s 1 1 + 18, 7 + 9s + 7, 9 + 9 s 1 + 1, 7 + 9s + 6) whee G 1 = Q0, 9 1 + 9 s + 9, ) + 1 1 Q5i, 9 1 + 9 s i + 7, 8) ) + Q5 1, 7 1 + 9 s + 10, 5), G = 1 P 5 1 + i 3, 7 1 5i 3, 8) ), G 3 = s Q9 1 + 9 s + 5i + 11, 7 1 + 9s i +, 8) ), G = P 9 1 + 3 s + 16, 7 1 + 8s, 6) + s 1 P 9 1 + 3 s + i + 15, 7 1 + 8s 5i 1, 8) ) + P 9 1 + 7 s + 15, 7 1 + 7 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 = 7 1 + 33, 9 1 + 18, 7 1 + 3, 9 1 + 19, 7 1 1 1 1 1 1 + 31, 9 + 0, 7 + 8, 9 + 1, 7 + 7, 9 +, 7 1 1 1 + 6, 9 + 3, 7 + 33). Othewise, let C = G 1 + 7 1 + 9 13

s 3 1 1 1 +17, 5, 7 +, 5 +1)+G +9 1 1 s 3 +1, 9 +9 +17) and C s = G 3 + G + 9 1 + 7 s 3 + 3, 7 1 + 9s + 6) whee G 1 = 1 G = 1 G 3 = Q9 1 G = s 3 Q5i 5, 9 1 i + 13, 8) ), P 5 1 + i 3, 7 1 5i 3, 8) ), + 18, 7 1 + 9s, 6) + s 3 1 Q9 1 + 5i + 17, 7 1 + 9s i 1, 8) ) + Q9 1 + 3 s 3 + 17, 7 1 + 8s + 3, 7), P 9 1 + 3 s 3 + i + 16, 7 1 + 8s 5i + 1, 8) ) + P 9 1 + 7 s 3 + 0, 7 1 + 7 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 + 31, 9 + 18, 7 + 30, 9 + 19, 7 + 7, 9 + 0, 7 + 6, 9 + 1, 7 + 31). Othewise, let C = G 1 + G + 9 + 3, 9 + 9 s + 17) and C s = G 3 + G + 9 + 7 s + 1, 7 + 9s + 13) whee G 1 = Q5i 5, 9 + 9 s i + 13, 8) ) + Q5, 7 + 9 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 9 + 3 s + i + 15, 7 + 8s 5i + 8, 8) ) + P 9 + 7 s + 15, 7 + 7 + 7, ). 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 + 3, 9 + ) and C s = G 3 + 9 + 3 s 3 + 5, 7 + 8s + 1, 9 + 3 s 3 + 1

6) + G + 9 + 7 s 3 + 8, 7 + 9s + 13) whee G 1 = Q0, 9 + 18, ) + 1 Q5i, 9 i + 16, 8) ) + Q5, 7 + 17, 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 9 + 3 s 3 + i +, 7 + 8s 5i + 7, 8) ) + P 9 + 7 s 3 + 6, 7 + 7 s 3 + 30, ). 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 + 9 3 + 5, 9 3 + 6) and C s = G 3 + G + 9 3 + 7 s 3 + 3, 7 3 + 7 s 3 + 0, 9 3 + 7 s 3 + 33, 7 3 9s + 0, 9 3 + 7, 7 3 + 9s + 19) whee G 1 = 3 Q5i 5, 9 3 i +, 8) ) + Q5 3, 7 3 + 1, 5), G = P 5 3 +, 7 3 +, 6) G 3 = s 3 P 5 3 + i + 1, 7 3 5i + 1, 8) ), Q9 3 + 5i +, 7 3 + 9s i + 15, 8) ) + 3 + Q9 3 + 3 s 3 + 9, 7 3 + 8s + 19, 3), G = P 9 3 + 3 s 3 + 30, 7 3 + 8s + 15, ) + s 3 P 9 3 + 3 s 3 + i + 8, 7 3 + 8s 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 = 15 1 + 17, 5 + 5, 15 1 + 1, 5 + 6, 15 1 + 17). 15

Othewise, let C = G 1 + + 5s, 3 1 1,, 3 + 1) + G + 5 1 + 1, 5 + 5s 1) and C s = G 3 + 15 1 s 1 + 9s + 9, 5 + 13 +, 15 1 s 1 + 9s + 8, 5 + 13 + 5) + G + 5 + 15 s 1 1 s 1 + 5, 15 + 15 + 1, 5 + 15 s 1 + 6, 15 1 + 10s + 7, 5 + 5s, 15 1 + 10s + 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, 15 1 + 10s i +, ), G = s 1 s 1 P 5 + 13 + i + 3, 15 1 + 9s 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 1 +1, 5+5s 1) and C s = G 3 + 15 1 + 9s + 8, 5 + 13 s 1 1, 15 + 9s + 6, 5 + 13 s ) + G + 5 + 15 s 1, 15 + 15 s s 1 + 7, 5 + 15 1, 15 + 10s + 7, 5 + 5s, 15 1 + 10s + 6) whee G 1 = 1 Q3i 3, 5 + 5s i 3, ), G = 1 G 3 = s 1 G = s 1 P 3 1 + i 1, 3i 5, ), Q5 + 5s + 3i 1, 15 1 + 10s i +, ), s 1 P 5 + 13 + i, 15 + 9s 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 3 + 15 s 1 s 1 + 9s, 5 + 13 + 5, 15 + 9s, 5 + 13 + 6) + G + 5 + 15 s 1 + 6, 15 + 10s 1) whee G 1 = 1 Q3i 3, 5 + 5s i 3, ), G = P 3 + i, 3i, ), G 3 = s 1 Q5 + 5s + 3i 3, 15 + 10s i 3, ), G = s 1 s 1 P 5 + 13 + i +, 15 + 9s 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

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 3 + 15 s + 9s + 1, 5 + 13 3, 15 + 9s, 5 + 13 ) + G + 5 + 15 s, 15 + 10s 1) whee G 1 = 1 Q3i 3, 5 + 5s i 3, ), G = P 3 + i, 3i, ), G 3 = s 1 Q5 + 5s + 3i 3, 15 + 10s i 3, ), G = s s P 5 + 13 + i, 15 + 9s 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 1 + 5 + 6s, ) + G + 3, 6 + 6s 1) and C s = G 3 + 9 + 11s 1, 6 + 8s + 1) + G + 6 + 9s, 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, 9 + 11s 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

Case 1.1: s. Let C = G 1 + G + 1, ) and C s+ = G 3 + G + G 5 + + 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 5 + 8 + 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 1 + 9 + 5s +, 13 1 + 5s + 9, 9 + 5s +, 13 1 + 5s + 10) + G + 15 1 + 5s + 10, 10 + 5s + 3) whee G 1 = 1 Q5 + 5s + 3i + 1, 10 + 5s i + 1, ), G = 1 1 P 13 + 5s + 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 ) 10 + 5s + 3. 18

Case.: is even. Let C = G 1 + 9 + 5s + 5, 13 + 5s + 1, 9 + 5s +, 13 + 5s + ) + G + 15 + 5s +, 10 + 5s + 3) whee G 1 = 1 Q5 + 5s + 3i + 1, 10 + 5s i + 1, ), G = P 13 + 5s + 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 ) 10 + 5s + 3. Case.3: s is odd. Let C s+ = G 3 + 5 + 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 3 + 5 + 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

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 5 + 8 + 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 + 6 + 5, +, 8 + 7) and C s+ = G 3 + 9 + 7, 11 + 10) + G + 10 + 9, 1 + 13, 8 + 8) whee G 1 = Q0, 6 + 6, + 1), G = P, 5 + 5, 1), G 3 = P 8 + 8, 10 + 1, ), G = Q9 + 9, 9 + 9, + 1). 0

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 < 3 + 1 and + s is odd. Let C + = G 1 + G + G 3 + + s + 5, + 1, + s + 7) and C s+ = G + G 5 + G 6 + G 7 + + 6s + 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 < 3 + 1 and + s is even. Let C + = G 1 + G + G 3 + + 1, + s + 7) and C s+ = G + G 5 + G 6 + G 7 + + 6s + 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 = 3 + 1. Let C + = G 1 + G + 1, 1 + 9, + 1, 16 + 11) and C s+ = G 3 + G + G 5 + G 6 + + 16, 8 + 3, 16 + 1) whee G 1 = Q0, 1 + 10, + 1), G = P, 13 + 11, ), G 3 = P 16 + 1, + 18, + 1), G = Q18 + 1, + 1, ), G 5 = Q19 + 1, 3 + 17, + 3), G 6 = P 0 + 15, 0 + 15, + ). 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

Case.5: s > 3 + 1 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 7 + + 6s + 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 > 3 + 1 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 7 + + 6s + 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.

If > 1, let C + = G 1 + G + 9 1 + 1, 9 +s 9 1, 9 1 3, 9 +s + ) whee G 1 = Q0, 9 +s 1, ) + 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 5 1 + i 3, 7 +s + s 5i, 8) ). +6, 9 +s 9 3 + 3 +s, 9 +8, 9 +) 3, ) + Q5i, 9 +s i, 8) ) + Q5 3 + 3, 7 +s + s +, 3), G = P 5 3 +, 7 +s + s, ) + 3 P 5 3 + i +, 7 +s + s 5i, 8) ). Case 3.1.3: s 1 mod ). If s = 1, let C s+ = 9 1 + 1, 9 1 + 19, 9 1 + 16, 9 1 + 18, 9 1 + 17, 9 1 + 1, 9 1 + 1). 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 3 + 10, 7) + 5 s 3 P 9 +s + 5 s 3 + i + 9, 9 +s + 7 s 3 5i + 9, 8) ). 3

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 + 1 + 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 1 + 7 +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 3 + 9 +s + 7 s +s + 7, 9 + 5 s + 6) + G + 9 +s 9 s +s + 6, 9 + 9 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 +, 9 + 7 s +, 6), + Q9 +s Case 3..: s mod ). Let C s+ = G 3 +9 +s P 9 +s + 5 s +s + i +, 9 + 7 s 5i +, 8)). +7 s +15, 9 +s +5 s + +s s +7, 9 +7 +1)+

G +9 +s whee +9 s +11, 9 +s +9 s +17, 9 +s +s s +5, 9 +9 +15) 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 1 9 1 1 +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 5 1 + 1, 7 +s 1 + s, ) + 1 1 P 5 1 + i 1, 7 +s 1 + s 5i 1, 8) ) + P 9 1 1, 9 +s 1 9 1 +, 5). Case 3.3.: 3 mod ). Let C + = G 1 + G + 9 +s 1 9 3, 9 3 + 7, 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 5 3 + i 1, 7 +s 1 + s 5i + 1, 8) ) + P 9 3 + 3, 9 +s 1 9 3 3, 5). Case 3.3.3: s 0 mod ). Let C s+ = G 3 + 9 +s 1 + 7 s +s 1 + 11, 9 + 5 s + 10) + G + 9 5

+s 1 + 9 s whee G 3 = s 1 G = s +s 1 + 10, 9 + 9 s +s 1 + 1, 9 + 9, 9 +s 1 + 9 s + 10) Q9 +s 1 + 5i + 6, 9 +s 1 + 9 s i + 6, 8)) + 5 s +s 1 + 6, 9 + 7 s + 8, 6), + Q9 +s 1 P 9 +s 1 + 5 s +s 1 + i + 6, 9 + 7 s 5i + 6, 8)). Case 3.3.: s mod ). Let C s+ = G 3 +9 +s 1 +7 s +s 1 +19, 9 +5 s +s 1 +11, 9 + 7 s + 18) + G + 9 +s 1 + 9 s + 15, 9 +s 1 + 9 s + 1, 9 +s 1 + 9, 9 +s 1 + 9 s + 19) whee G 3 = s Q9 +s 1 + 5i + 6, 9 +s 1 + 9 s i + 15, 8) ), G = Q9 +s 1 + 5 s + 13, 9 +s 1 + 7 s + s + 13, 5) P 9 +s 1 + 5 s + i + 11, 9 +s 1 + 7 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 + 5 1 + 1, 5 1 + 5s + 5, 5 1 + 3, 5 + 5s + ) whee G 1 = 1 G = 1 1 Q3i 3, 5 + 5s i +, )) + Q3, + 5s +, 3), P 3 1 + i 1, + 5s 3i, ) ). Case.: is even. Let C + = G 1 + + 5s +, 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+1 +5 1, 5+10s+8, 5+5s+5, 5+10s+6) 6

whee G 3 = s 1 Q5 + 5s + 3i +, 5 + 10s i +, )), G = Q13 s+1 + 5, 5 + 9s +, 3) + s 1 P 13 s+1 + 5 + i 1, 5 + 9s 3i + 3, ) ). Case.: s is even. Let C s+ = G 3 + 5 + 9s + 8, 13 s s + 5 +, 5 + 9s + 7, 13 + 5 + 6) + G + 15 s + 5 + 6, 5 + 10s + 8, 5 + 5s + 5, 5 + 10s + 6) whee G 3 = s 1 G = s Q5 + 5s + 3i +, 5 + 10s i +, )), P 13 s + 5 + 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 1 + 5 + 6s + 5, ) + G + 3 1, 3 + 6s + 3, 3 + 1, 6 + 6s + 5) and C s+ = G 3 + 6 + 11s + 9, 6 + 8s + 5) + G + 6 + 9s + 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 +, 6 + 11s 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

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. 150 1996), 3 15. [] P. Adams, D. Byant, and M. Buchanan, A suvey on the existence of G-designs, J. Combin. Des. 16 008), 373 10. [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), 77 8. [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. 77 85. [6] M. Buatti, Recusive constuctions fo diffeence matices and elative diffeence families, J. Combin. Des. 6 1998), 165 18. [7] M. Buatti and L. Gionfiddo, Stong diffeence families ove abitay gaphs, J. Combin. Des. 16 008), 3 61. [8] M. Buatti and A. Pasotti, Gaph decompositions with the use of diffeence matices, Bull. Inst. Combin. Appl. 7 006), 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), 09 19. [10] S. I. El-Zanati and C. Vanden Eynden, On Rosa-type labelings and cyclic gaph decompositions, Mathematica Slovaca 59 009), 1 18. [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. 55 60. 8

[13] A. Pasotti, On d-gaceful labelings, As Combin. 111 013), 07 3. [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. 39 355. [15] D. Sotteau, Decomposition of K m,n K m,n) into cycles cicuits) of length k, J. Combin. Theoy, Se. B, 30 1981), 75 81. 9