A note on cyclic m-cycle systems of K-r((m))

Digital Object Identifier (DOI) 10.1007/s00373-006-0658-z

Graphs and

Combinatorics

A Note on Cyclic m-cycle Systems of K

Shung-Liang Wu1_{and Hung-Lin Fu}2

1 _{National United University, Miaoli, Taiwan, R.O.C.}

2_{Department of Applied Mathematics, National Chaio Tung University, Hsin Chu, Taiwan,}

R.O.C.

Abstract. It was proved by Buratti and Del Fra that for each pair of odd integers r and m, there exists a cyclic m-cycle system of the balanced complete r-partite graph Kr(m)except for

the case when r = m = 3. In this note, we study the existence of a cyclic m-cycle system of

Kr(m)where r or m is even. Combining the work of Buratti and Del Fra, we prove that cyclic m-cycle systems of Kr(m)exist if and only if (a) Kr(m)is an even graph (b) (r, m)
= (3, 3) and

(c) (r, m)
≡ (t , 2) (mod 4) where t ∈ {2, 3}.

The Main Result

An m-cycle system of a simple graph G is a set C of edge disjoint m-cycles which partition the edge set of G. The necessary conditions for the existence of an m-cycle system of a graph G are that the value of m is not exceeding the order of G, m divides the number of edges in G, and the degree of each vertex in G is even. An even graph is a graph with vertex degrees all even. If G is a complete graph Kvon v vertices,

then it is called an m-cycle system of order v.

Alspach and Gavlas [1] and ˇSajna [11] have completely settled the existence prob-lem of m-cycle systems of Kv and Kv− I, where I is a 1-factor. Moreover, there

have been many results in the literature concerning the existence problem of cyclic

m-cycle systems. The reader can refer to [2–10, 12, 13].

A graph G is said to be a complete r-partite graph (r > 1) if its vertex set V can be partitioned into r disjoint non-empty sets V1, . . . , Vr (called partite sets) such

that there exists exactly one edge between each pair of vertices from different partite sets. If|V_i| = n_ifor 1≤ i ≤ r, the complete r-partite graph is denoted by K_n1,···,nr. In particular, if n1= · · · = nr = k(> 1), it is called balanced and the graph will be

simply denoted by Kr(k).

Let C = (c0, . . . , cm−1) be an m-cycle. An m-cycle system of a graph G, C, is

said to be cyclic if V (G) = Zvand (c0+ 1, . . . , cm−1+ 1) ∈ C (mod v) whenever

(c0, . . . , cm−1) ∈ C.

The necessary conditions for the existence of a cyclic m-cycle system of a com-plete r-partite graph G are that G is balanced, say Kr(k), and any partite set in Kr(k)

rZk = {0, r, . . . , (k − 1)r}

of Zrkor its coset. For i = 0, . . . , r − 1, let Vi denote the ith partite set of Kr(k).

We may assume Vi = {i, i + r, i + 2r, . . . , i + (k − 1)r}. Note that the set of distinct

differences of edges in Kr(k)is Zrk\ ± {0, r, 2r, . . . , k/2 r}.

For any edge{a, b} in G with V (G) = Z_v, we shall use± |a − b| to denote the

difference of the edge{a, b}. The number of distinct differences in a cycle C is called

the weight of C.

Let m = ab be a positive integer (> 2). An m-cycle C in Kr(k)with weight a has

index rk

b if for each edge{s, t} in C, the edges {s + i ·rkb, t + i ·rkb} ( mod rk) with

i ∈ Zbare also in C.

Proposition 1 ([14]). Let m = ab be a positive integer (> 2). Then there exists an

m-cycle C = (c0, . . . , cm−1) in Kr(k)with weight a and indexrk_b if and only if each of

the following conditions is satisfied:

(1) for 0≤ i
= j ≤ a − 1, c_i
≡ c_j(mod rk

b);

(2) the differences of the edges{c_i, ci−1} (1 ≤ i ≤ a) are all distinct;

(3) ca= c0+ t ·rk_b, where gcd (t, b) = 1; and

(4) cia+j = cj+ i · t · rk_b where 0≤ j ≤ a − 1 and 0 ≤ i ≤ b − 1.

To simplify, the m-cycle C = (c0, . . . , ca−1, c0+ t ·rk_b, . . . , ca−1+ t ·rk_b, . . . , c0+

(b − 1) · t ·rk_b, . . . , ca−1+ (b − 1) · t ·rk_b) will be denoted by C = [c0, . . . , ca−1]t·rk/b.

Note that if C is any cycle with weight a in a cyclic m-cycle system of Kr(k), then

C is precisely an m-cycle with indexrk_b.

The following results are either known or easy to verify, we list them without the details of proof.

Theorem 2 ([2]). For each pair of odd integers r and m, there exists a cyclic m-cycle

system of Kr(m)with the exception that (r, m) = (3, 3).

Lemma 3 ([14]). If there is a cyclic m-cycle system of a graph G, then G is 2r-regular

for some positive integer r.

Proposition 4 ([14]). If there is a cyclic m-cycle system of Kr(m)with m even and m > 4, then (r, m)
≡ (t, 2) (mod 4) where t ∈ {2, 3}.

Note that if m is odd, then r must be odd since Kr(m)is an even graph.

For an m-cycle C = (c0, . . . , cm−1), we shall use ∂C to denote the set of distinct

differences{±(c_i−c_i−1)|i = 1, . . . , m} where cm= c0. Given a set D = {C1, . . . , Cp}

of m-cycles, the list of differences from D is defined as the union of the multisets

∂C1, . . . , ∂Cp, i.e., ∂D =

_p

i=1∂Ci.

Theorem 5 ([14]). Let D be a set of m-cycles with vertices in Zrk such that ∂D = Zrk\ ± {0, r, 2r, . . . , k/2 r}. Then there exists a cyclic m-cycle system of Kr(k).

We are now ready for the main result. First, we will assume Ci = (vi,0, vi,1, . . . ,

vi,2s, vi,2s+1, vi,2s, vi,2s−1, . . . , vi,1) to be a (4s + 2)-cycle, and an m-cycle with

weight m is called full, otherwise short.

Theorem 6. A cyclic m-cycle system of Kr(m)exists if and only if (a) Kr(m)is an even

graph (b) (r, m)
= (3, 3) and (c) (r, m)
≡ (t, 2) (mod 4) where t ∈ {2, 3}.

Proof. The necessary part follows by Theorem 2 and Proposition 4. Therefore, we

prove the sufficiency in what follows. The proof is split into 4 cases: (i) (r, m) ≡ (0, 2) (mod 4) (ii) (r, m) ≡ (1, 2) (mod 4) (iii) r ≡ 0 (mod 2) and m ≡ 0 (mod 4) (iv) r ≡ 1 (mod 2) and m ≡ 0 (mod 4). Note that if m is odd, then r must be odd and this case has been settled by Buratti and Del Fra in [2].

Case 1. (r, m) ≡ (0, 2) (mod 4).

Subcase 1.1. r ≡ 0 (mod 4) and m ≡ 2 (mod 8), say r = 4p and m = 8k + 2.

Let C∗= [c0, . . . , c4k]r(4k+1)be a short m-cycle defined as

ci =          2rj, if i = 2j with j = 0, . . . , 2k − 1; 4r − 1 + 8j r, if i = 4j + 1 with j = 0, . . . , k − 1; 7r − 1 + 8j r, if i = 4j + 3 with j = 0, . . . , k − 2; r(8k − 1) + 1, if i = 4k − 1; and 4rk + 2, if i = 4k.

It can be checked that all values in C∗are certainly pairwise distinct and the set of differences occurring in C∗is ∂C∗= ±{r − 2, 2r − 1, 3r − 1, . . . , (4k + 1)r − 1}.

For i = 1, . . . , p, the full m-cycles Ciare defined as

vi,0 = 0; for j = 0, . . . , 2k − 1, vi,2j +1 = jr − 3 + 4i, vi,2j +1 = vi,2j +1+ 2;

for j = 1, . . . , 2k − 1, vi,2j = r(4k + 1 − j) − 6 + 8i, vi,2j = vi,2j + 3; vi,4k =

r(2k + 1) − 5 + 8i, vi,4k = vi,4k+ 14; and vi,4k+1= 2rk − 2 + 4i.

If p ≥ 2, then for i = 1, . . . , p − 1, the remaining full m -cycles Cp+iare given

by

vp+i,0 = 0; for j = 0, . . . , 2k − 1, vp+i,2j +1 = jr − 2 + 4i, vp+i,2j +1 =

vp+i,2j +1+ 2; for j = 1, . . . , 2k − 1, vp+i,2j = r(4k + 1 − j) − 3 + 8i, vp+i,2j =

vp+i,2j + 3; vp+i,4k = r(2k + 1) − 2 + 8i, vp+i,4k = vp+i,4k+ 1; and vp+i,4k+1 =

2rk − 1 + 4i.

By routine computation, we have that all values in each full m-cycle constructed above are also pairwise distinct and
2p−1_i=1 ∂Ci = ±{1, 2, . . . , r − 3, r − 1} ∪

_4k−1

i=0 ±{r + 1 + ir, r + 2 + ir, . . . , 2r − 2 + ir}.

Since ∂C∗∪
2p−1_i=1 ∂Ci = Zrm\ ± {0, r, 2r, . . . , rm/2}, it follows from Theorem

5 that there exists a cyclic m-cycle system of Kr(m).

If k = 0, then C∗ = [0, 4r − 1, 3r − 2]3r is the short 6-cycle. For i = 1, . . . , p,

the full 6-cycles are Ci = (0, 4i − 3, 2r − 4 + 8i, r − 2 + 4i, 2r − 3 + 8i, 4i − 1) and,

if p ≥ 2, for i = 1, . . . , p − 1, the remaining full 6-cycles are Cp+i = (0, 4i, 2r +

1+ 8i, r + 1 + 4i, 2r + 2 + 8i, 4i + 2).

We then have that ∂C∗ = ±{2, r + 1, 2r + 1} and
2p−1_i=1 ∂Ci = ±{1, 3, 4, . . . ,

r − 1, r + 2, r + 3, . . . , 2r − 1, 2r + 2, 2r + 3, . . . , 3r − 1}.

If k > 0, then the short m-cycle C∗= [c0, . . . , c4k+2]r(4k+3)is defined as

ci =          2j r, if i = 2j with j = 0, . . . , 2k; 3r + 1 + 8j r, if i = 4j + 1 with j = 0, . . . , k − 1; 6r + 1 + 8j r, if i = 4j + 3 with j = 0, . . . , k − 1; 4r(2k + 1) − 1, if i = 4k + 1; and r(4k + 3) − 2, if i = 4k + 2, and ∂C∗= ±{2, r + 1, 2r + 1, . . . , (4k + 2)r + 1}. For i = 1, . . . , p, the full m-cycles Ciare defined as

vi,0= 0; for j = 0, . . . , 2k, vi,2j +1= jr −3+4i, vi,2j +1 = vi,2j +1+2, vi,2j +2=

r(4k + 2 − j ) − 4 + 8i, and vi,2j +2 = vi,2j +2+ 1; and vi,4k+3 = r(2k + 1) − 2 + 4i.

For i = 1, . . . , p − 1, the rest of full m-cycles Cp+iare given by

vp+i,0 = 0; for j = 0, . . . , 2k, vp+i,2j +1= jr + 4i, vp+i,2j +1 = vp+i,2j +1+ 2,

vp+i,2j +2= r(4k + 2 − j) + 1 + 8i, and vp+i,2j +2 = vp+i,2j +2+ 1; and vp+i,4k+3=

r(2k + 1) + 1 + 4i.

An easy verification shows that
2p−1_i=1 ∂Ci = ±{1, 3, 4, . . . , r−1}∪

_4k+1

i=0 ±{(r+

2+ ir, r + 3 + ir, . . . , 2r − 1 + ir}.

Case 2. r ≡ 1 ( mod 4) and m ≡ 2 (mod 4), say r = 4p + 1 and m = 4k + 2.

It suffices to consider the full m- cycles.

For i = 1, . . . , p, the full m-cycles Ciare defined as

vi,0 = 0; for j = 0, . . . , k − 1, vi,2j +1 = jr − 3 + 4i, vi,2j +1 = vi,2j +1+ 2;

for j = 1, . . . , k − 1 (if k ≥ 2), vi,2j = r(2k + 1 − j) − 6 + 8i, vi,2j = vi,2j+ 3;

vi,2k = r(k + 1) − 5 + 8i, vi,2k = vi,2k+ 1; and vi,2k+1= rk − 2 + 4i.

We have
p_i=1∂Ci = ±{1, 3, . . . , r −2}∪

_p−1

i=0

_2k

j =1±{jr +1+4i, jr +2+4i}.

For i = 1, . . . , p, let Cp+ibe the rest of the full m-cycles given by

vp+i,0 = 0; for j = 0, . . . , k−1, vp+i,2j +1= jr−2+4i, vp+i,2j +1 = vp+i,2j +1+

2; for j = 1, . . . , k − 1, vp+i,2j = r(2k + 1 − j) − 3 + 8i, vp+i,2j = vp+i,2j + 3;

vp+i,2k= r(k + 1) − 2 + 8i, vp+i,2k = vp+i,2k+ 1; and vp+i,2k+1= rk − 1 + 4i.

It follows that
p_i=1∂Cp+i = ±{2, 4, . . . , r −1}∪

_p−1

i=0

_2k

j =1±{jr +3+4i, jr +

4+ 4i}, and
2p_i=1∂Ci = Zrm\ ± {0, r, 2r, . . . , (2k + 1)r}.

Case 3. r ≡ 0 (mod 2) and m ≡ 0 (mod 4). Subcase 3.1. m ≡ 0 (mod 8), say m = 8k.

For i = 1, . . . , r − 1, the short m-cycles C_i∗= [ci,0, . . . , ci,4k−1]4rkare defined as ci,j =    2rs, if j = 2s with s = 0, . . . , 2k − 1; 2r + i + 8rs, if j = 4s + 1 with s = 0, . . . , k − 1; and 5r + i + 8rs, if j = 4s + 3 with s = 0, . . . , k − 1.

We have ∂C_i∗ = ±{i, r + i, 2r + i, . . . , (4k − 1)r + i} and
r−1_i=1∂C∗_i = Zrm\ ±

{0, r, 2r, . . . , 4rk}.

Subcase 3.2. m ≡ 4 (mod 8), say m = 8k + 4.

For i = 1, . . . , r − 1, the short m-cycles C_i∗= [ci,0, . . . , ci,4k+1]r(4k+2)are given by

ci,j =    2rs, if j = 2s with s = 0, . . . , 2k; r + i + 8rs, if j = 4s + 1 with s = 0, . . . , k; and 6r + i + 8rs, if k ≥ 1 and j = 4s + 3 with s = 0, . . . , k − 1. ∂C_i∗ = ±{r − i, r + i, 2r + i, 3r + i, . . . , (4k + 1)r + i} and
r−1_i=1∂C_i∗ = Zrm\ ± {0, r, 2r, . . . , (4k + 2)r}. Case 4. (r, m) ≡ (t, 0) (mod 4), t ∈ {1, 3}. Subcase 4.1. m ≡ 4 (mod 8), say m = 8k + 4.

For i = 1, . . . , r − 1, the short m-cycles are C_i∗ = [0, i]2r and

_r−1

i=1∂Ci∗ =

±{1, 2, . . . , r − 1, r + 1, r + 2, . . . , 2r − 1}.

If k ≥ 1 then for i = 1, . . . , r −1, and j = 1, . . . , k , the remaining short m-cycles are C_i,j∗ = [0, 4jr + i]2r and ˆC_i,j∗ = [0, (4j + 1)r + i]2r.

By routine computation, it follows that
r−1_i=1
k_{j =1}∂C_i,j∗ =
k−1_s=0±{2r + 1 +

4sr, 2r + 2 + 4sr, . . . , 3r − 1 + 4sr, 4r + 1 + 4sr, 4r + 2 + 4sr, . . . , 5r − 1 + 4sr} and
_r−1 i=1
_k j =1∂ ˆCi,j∗ =
_k−1 s=0±{3r + 1 + 4sr, 3r + 2 + 4sr, . . . , 4r − 1 + 4sr, 5r + 1 + 4sr, 5r + 2 + 4sr, . . . , 6r − 1 + 4sr}.

Subcase 4.2. m ≡ 0 (mod 8), say m = 8k.

For i = 1, . . . , r − 1, andj = 1, . . . , k, the short m -cycles are C_i,j∗ = [0, (4j − 2)r + i]2rand ˆCi,j∗ = [0, (4j − 1)r + i]2r.

We have
r−1_i=1
k_{j =1}∂C_i,j∗ =
k−1_s=0±{1 + 4sr, 2 + 4sr, . . . , r − 1 + 4sr, 2r + 1 +

4sr, 2r + 2 + 4sr, . . . , 3r − 1 + 4sr} and
r−1_i=1
k_{j =1}∂ ˆC_i,j∗ =
k−1_s=0±{r +1+4sr, r +

2+ 4sr, . . . , 2r − 1 + 4sr, 3r + 1 + 4sr, 3r + 2 + 4sr, . . . , 4r − 1 + 4sr}.
We end this note with a conclusion. Assume m to be even (> 2) and Km− I

to be the complete graph with 1-factor I removed. Observing the consequence of Theorem 6, it is clear that if there exists a cyclic m -cycle system of Km− I, then

a cyclic m-cycle system of Krm− I is given. Unfortunately, there does not exist a

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Received: December 16, 2004