Digital Object Identifier (DOI) 10.1007/s00373-008-0791-yGraphs and Combinatorics (2008) 24:367–371

Graphs andCombinatorics© Springer-Verlag 2008

Embedding Partial 4-cycle Systems of Arbitrary Index

April Parker, C. A. RodgerDepartment of Mathematics and Statistics, 221 Parker Hall, Auburn University,AL 36849-5310, e-mail: parkera@troy.edu, e-mail: rodgec1@auburn.edu

Abstract. Recently, Lindner showed that every partial 4-cycle system of order n and index1 could be embedded in a 4-cycle system of order v and index 1 with v ≤ 2n + 15. Whilethe technique he used does not immediately extend to any higher index, here we develop hisideas to show that every partial 4-cycle system of order n and index λ can be embedded in a4-cycle system of order v and index λ for all λ-admissible v ≥ 8�n/4� + 7, v �= 8�n/4� + 8.This improves on the best known bounds of v = 4n and v = 8n + 1 when λ > 1 is even andodd respectively.

1. Introduction

Over the past 35 years various results concerning the embedding of partial graphdesigns have appeared in journals. A G-design of H is a partition of E(H) intosets, each of which induces a copy of G. A G-design of (a subgraph of) λKn , thegraph with λ-edges between each pair of the n vertices, is also known as a (partial)G-system of order n and index λ; it is often denoted by (V, B) where V is the vertexset of λKn and B is a collection of copies of G whose edges form a partition of (asubgraph) of E(λKn). A partial G-system of order n and index λ (V1, B1) is said tobe embedded in a G-system of order v and index λ (V2, B2) if V1 ⊆ V2 and B1 ⊆ B2.

Typically one is interested in finding the values of v for which each partialG-system of order n and index λ can be embedded in some G-system of orderv. The history of embeddings of partial cycle systems is particularly rich. In 1973,Lindner conjectured that every partial 3-cycle system of order n and index λ couldbe embedded in a 3-cycle system of order v for all λ-admissible v ≥ 2n + 1 [6].(v is said to be λ-admissible if each vertex in λKv has even degree and |E(λKv)|is divisible by the number of edges in Ck , both conditions being necessary for theexistence of such a cycle system). This result was proved true when 4 divides λ in1991 by Hilton and Rodger [4], when λ is even in 1994 by Johansson [5] (see also[3]), and most recently when λ = 1 by Bryant and Horsley [1]. The lower boundv ≥ 2n + 1 is best possible; indeed it can be shown [9] that if k is odd then to embedeach partial Ck-system of order n in some Ck-system of order v it is necessary thatv ≥ (k +1)n/(k −1)+1. We conjecture that this is also sufficient. For even cycles, itis not so clear what this lower bound on v should be. The best results to date can befound in [2]. In particular, it has been shown that every partial C4-system of order

368 A. Parker, C. A. Rodger

n and index λ can be embedded in a C4-system of order v where v ≤ 2n + 15 whenλ = 1 [7], v = 4n when λ > 1 is even [8], and v = 8n + 1 when λ > 1 is odd [8].

The technique used by Lindner in the case where λ = 1 does not immediatelyextend to higher values of λ, but here we develop his ideas to show that every C4-design of order n and index λ can be embedded in a C4-design of order v for allλ-admissible v ≥ 8�n/4�+7, v �= 8�n/4�+8, thus improving upon previous boundsby a factor of approximately 2 and 4 if λ is even or odd respectively.

2. Preliminary Results

We begin with some C4-designs of G for some particular choices of G in whichcertain 4-cycles are guaranteed to occur. These are building blocks used in the mainconstruction.

It is easy to show that there exists a 4-cycle system of the complete λ-fold bipar-tite graph λKx,y if and only if x, y ≥ 2, each vertex has even degree, and the numberof edges is divisible by 4 (see Sotteau [10] for a more general result). Throughout thispaper, we denote such a set of 4-cycles by B(X, Y ), where {X, Y } is the bipartitionof the vertex set of λKx,y .

The following result is a slight adaptation of the observation by Lindner usedin his construction [7].

Lemma 1. Let m ≡ 0 (mod 4). There exists a 4-cycle system (W, B3) of G ∨ K c7 with

W = (Zm × Z2) ∪ {∞i |i ∈ Z7}, where G consists of m/2 vertex disjoint copies of K4(on the vertex sets {2i, 2i + 1} × {0, 1}, i ∈ Zm/2), in which

((2i, 0), (2i + 1, 0), (2i, 1),∞0) ∈ B3 for each i ∈ Zm/2.

Proof. The result follows by defining

B3 = {((2i, 0), (2i + 1, 0), (2i, 1),∞0), ((2i, 0), (2i, 1),∞1, (2i + 1, 1))

((2i + 1, 0), (2i + 1, 1), (2i, 1),∞2) | i ∈ Zm/2}∪B({(2i, 1) | i ∈ Zm/2}, {∞ j | j ∈ Z7\Z3})∪B({(2i, 0) | i ∈ Zm/2}, {∞ j | j ∈ Z7\{0}})∪B({(2i + 1, 0) | i ∈ Zm/2}, {∞ j | j ∈ Z7\{2}})∪B({(2i + 1, 1) | i ∈ Zm/2}, {∞ j | j ∈ Z7\{1}}).

�

Lemma 2. There exist 4-cycle systems (W, B1(a, b)) and (W, B2(a, b)) of 2(K4 ∨ K c7)

with W = ({a, b} × {0, 1}) ∪ {∞i | i ∈ Z7} in which

(1) B1(a, b) contains two copies of ((a, 0), (b, 0), (a, 1),∞0), and(2) B2(a, b) contains ((a, 0), (b, 0), (a, 1),∞0) and ((b, 0), (a, 0), (b, 1),∞0).

Proof. Let B1 = {((a, 0), (b, 0), (a, 1),∞0), ((a, 0), (b, 0), (a, 1),∞0),((a, 0), (a, 1),∞1, (b, 1)), ((a, 0), (a, 1),∞1, (b, 1)), ((b, 0), (b, 1), (a, 1),∞2),((b, 0), (b, 1), (a, 1),∞2), ((b, 0),∞5, (a, 1),∞6), ((b, 0),∞0, (b, 1),∞3),

Embedding Partial 4-cycle Systems of Arbitrary Index 369

((a, 0),∞2, (b, 1),∞5), ((b, 0),∞1, (a, 0),∞6), ((b, 0),∞3, (a, 1),∞5),((b, 1),∞4, (a, 1),∞6), ((a, 0),∞3, (a, 1),∞4), ((a, 0),∞3, (b, 1),∞5),((b, 0),∞1, (a, 0),∞4), ((a, 0),∞2, (b, 1),∞6), ((b, 0),∞0, (b, 1),∞4)}.

Let B2 = {((a, 0), (b, 0), (a, 1),∞0), ((b, 0), (a, 0), (b, 1),∞0),((a, 0), (a, 1),∞1, (b, 1)), ((b, 0), (b, 1),∞1, (a, 1)), ((b, 0), (b, 1), (a, 1),∞2),((a, 0), (a, 1), (b, 1),∞2), ((b, 0),∞5, (a, 0),∞6), ((b, 0),∞1, (a, 0),∞4),((b, 0),∞1, (a, 0),∞3), ((b, 0),∞0, (a, 1),∞3), ((b, 0),∞2, (a, 1),∞4),((b, 1),∞5, (a, 1),∞6), ((b, 0),∞5, (a, 0),∞6), ((b, 1),∞3, (a, 1),∞4),((a, 0),∞0, (b, 1),∞3), ((a, 0),∞2, (b, 1),∞4), ((b, 1),∞5, (a, 1),∞6)}. �

3. The Main Construction

We are now ready to provide the new embeddings.

Theorem 1. Every partial 4-cycle system of order n and index λ can be embedded in a4-cycle system of order v and index λ for all λ-admissible v ≥ 8�n/4� + 7, v �=8�n/4� + 8.

Proof. Let m = 4�n/4� and let w = v − 2m. Note that by the condition of thetheorem, w ≥ 7 and w �= 8. So, since v is λ-admissible, since w ≥ 7 and since w ≡ v

(mod 8), it follows that w is λ-admissible and so there exists a 4-cycle system ofλKw [2]. The fact that v and w are both λ-admissible also means either that w andv are both odd, or that λ is even; so since v − w �= 1 it follows from the discussionat the beginning of Section 2 that there exists a 4-cycle system of λK2m,v−w. Theseingredients will be used in the construction.

We define the desired 4-cycle system on the vertex set V = (Zm × Z2) ∪ {∞i |i ∈ Zw}. Let (Zm, B) be the given partial 4-cycle system (adding up to 3 isolatedvertices if necessary). Form B ′ from B by arbitrarily orienting the edges so that each4-cycle in B becomes a directed 4-cycle in B ′.

For each (a, b) ∈ {(2i, 2i + 1) | i ∈ Zm/2} = H , let m′(a, b) and m(b, a) bethe number of arcs directed from a to b and from b to a respectively; by renamingvertices if necessary we can assume that m′(a, b) ≥ m(b, a). Since B has index λ,m′(a, b) + m(b, a) ≤ λ for all (a, b) ∈ H , so by defining m(a, b) = λ − m(b, a), wehave:

(i) m(a, b) ≥ m′(a, b) ≥ m(b, a), and(ii) m(a, b) + m(b, a) = λ.

The cases when λ is even and λ is odd are considered separately, though they havemany features in common, as will be described.

Case 1: λ is even Construct the desired 4-cycle system (V, F) as follows (the follow-ing will also be used when λ is odd, hence the use of λ/2�, despite this being thecase when λ is even):

370 A. Parker, C. A. Rodger

(1) Let ({∞i |i ∈ Zw}, F1) be a 4-cycle system of index λ and let F1 ⊂ F .(2) For each (a, b) ∈ H , let F contain m(b, a) copies of the 4-cycles in B2(a, b)

defined in Lemma 2, and let F contain λ/2� − m(b, a) copies of the 4-cyclesin B1(a, b).

(3) Let (V, F2) be a 4-cycle system of λK2m,w−7 with bipartition {{Zm × Z2},{∞i |i ∈ Zw\Z7}} of V , and let F2 ⊆ F .

(4) For 0 ≤ a < b < m with (a, b) /∈ H , let F contain λ copies of ((a, 0), (b, 0),

(a, 1), (b, 1)).

Then for each pair of vertices y, z ∈ V :

(a) if {y, z} ⊆ (Zm × Z2) and there is no (a, b) ∈ H for which {y, z} ⊆ {a, b} × Z2then λ 4-cycles defined in (4) contain the edge {y, z};

(b) if {y, z} ⊆ {a, b} × Z2 for some (a, b) ∈ H , or if y ∈ Zm × Z2 and z ∈ {∞i |i ∈Z7}, then 2(m(b, a) + �λ/2� − m(b, a)) = 2�λ/2� = λ 4-cycles defined in (2)contain the edge {y, z};

(c) if y ∈ Zm × Z2 and z ∈ {∞i |i ∈ Zw\Z7} then λ 4-cycles defined in (3) containthe edge {y, z}; and

(d) if {y, z} ⊆ {∞i |i ∈ Zw} then λ 4-cycles defined in (1) contain the edge {y, z}.So (V, F) is a 4-cycle system of order v and index λ. But as yet we may not have

a copy of B in F . For each 4-cycle (a, b, c, d) ∈ B, we now replace four 4-cycles inF with four mutually balanced 4-cycles, one of which is ((a, 0), (b, 0), (c, 0), (d, 0)).

Let ( j, k, s, t) be a directed 4-cycle in B ′; name the vertices so that if the directed4-cycle contains one or two arcs in H then the arcs are ( j, k), or ( j, k) and (s, t)respectively. If ( j, k, s, t) contains:

(a) 0 arcs in H then F contains the 4-cycles in {((a, 0), (b, 0), (a, 1), (b, 1))|{a, b} ∈ {{ j, k}, {k, s}, {s, t}, { j, t}} (see (4)) which are replaced by the 4-cyclesin {(( j, 0), (k, 0), (s, 0), (t, 0)), (( j, 0), (k, 1), (s, 0), (t, 1)),(( j, 1), (k, 0), (s, 0), (t, 1)), (( j, 1), (k, 1), (s, 1), (t, 1));

(b) 1 arc in H then F contains the 4-cycles in {(( j, 0), (k, 0), ( j, 1),∞0),((k, 0), (s, 0), (k, 1), (s, 1)), ((s, 0), (t, 0), (s, 1), (t, 1)),(( j, 0), (t, 0), ( j, 1), (t, 1))} (see (2) and (4)) which are replaced with the 4-cyclesin {(( j, 0), (k, 0), (s, 0), (t, 0)), (( j, 0), (t, 1), ( j, 1),∞0),(( j, 1), (k, 0), (s, 1), (t, 0)), ((k, 1), (s, 0), (t, 1), (s, 1))}; and

(c) 2 arcs in H then F contains the 4-cycles {(( j, 0), (k, 0), ( j, 1),∞0),((k, 0), (s, 0), (k, 1), (s, 1)), ((s, 0), (t, 0), (s, 1),∞0),(( j, 0), (t, 0), ( j, 1), (t, 1))} (see (2) and (4)) which are replaced with the 4-cyclesin {(( j, 0), (k, 0), (s, 0), (t, 0)), (( j, 0), (t, 1), ( j, 1),∞0),(( j, 1), (k, 0), (s, 1), (t, 0)), ((s, 0), (k, 1), (s, 1),∞0)}.

Clearly a copy of the given partial 4-cycle system now occurs on the vertex setZm × {0}. Notice that the definitions of m′(a, b) and of m(a, b) ensure that thereare enough 4-cycles of the required types in F in order to complete these replace-ments. To see this, using (i) and (ii), note that Lemma 2 guarantees that Step 2 ofthe construction produces

2(λ/2 − m(b, a)) + m(b, a) = λ − m(b, a) = m(a, b) ≥ m′(a, b)

Embedding Partial 4-cycle Systems of Arbitrary Index 371

copies of ((a, 0), (b, 0), (a, 1),∞0), and produces m(b, a) copies of ((b, 0), (a, 0),

(b, 1),∞0) as required in the replacement process.

Case 2: λ is odd Notice that when λ is odd, the construction used in the previouscase covers each edge λ times except in Step 2 where it is only covered 2 λ/2� times.Of course, when λ is even 2 λ/2� = λ, but when λ is odd 2 λ/2� = λ − 1; somore 4-cycles are needed. Also, for each (a, b) ∈ H , since m(a, b) = λ − m(b, a) =2 λ/2� + 1 − m(b, a), we need to cover these additional edges with 4-cycles in sucha way that they include one more copy of ((a, 0), (b, 0), (a, 1),∞). This is preciselywhat Lemma 1 does. Adding B3 to F therefore produces a 4-cycle system of λKv,and performing the replacements defined in (a − c) of the previous case producesan embedding of the given partial 4-cycle system. �

As a final comment, it is worth noting that one could tinker with this construc-tion to reduce the size of the embedding to something closer to 2n, making theconstruction longer and more complicated; but the aim now should be to reducethe size of the smallest embedding to v = cn + d where c < 2. This is expected torequire a completely new approach.

References

1. Bryant, D., Horsley, D.: A proof of Lindner’s Conjecture on embeddings of partial SteinerTriple Systems, submitted

2. Bryant, D., Rodger, C.A.: Cycle Decompositions, The CRC handbook of combinatorialdesigns, CRC Press Series on Discrete Mathematics and its Applications (second edition,C. J. Colbourn and J. H. Dinitz (eds),), CRC Press, Boca Raton, FL, 373–381 (2007)

3. Ferencak, M.N., Hilton, A.J.W.: Outline and amalgamated triple systems of even index,Proc. London Math. Soc. 84(3), 1–34 (2002)

4. Hilton, A.J.W., Rodger, C.A.: The embedding of partial triple systems when 4 divides λ,J. Combin. Theory (B) 56, 109–137 (1991)

5. Johansson, A.: Some Results on Colouring of Graphs, PhD Dissertation, University ofUmea, 1994

6. Lindner, C.C.: A partial Steiner triple system of order n can be embedded in a Steinertriple system of order 6n + 3. J. Combin. Theory (A) 18, 349–351 (1975)

7. Lindner, C.C.: Il primo amore non si scorda or an up-to-date survey of small embeddingsfor partial even-cycle systems. J. Geom. 76, 183–199 (2003)

8. Lindner, C.C., Rodger, C.A.: Embedding directed and undirected partial cycle systemsof index λ > 1, J. Combin. Designs 1, 113–123 (1993)

9. Rodger, C.A.: Problems on cycle systems of odd length. Seventeenth Manitoba Confer-ence on Numerical Mathematics and Computing. Congr. Numer. 61, 5–22 (1988)

10. Sotteau, D.: Decomposition of Km,n(K ∗m,n) into cycles (circuits) of length 2k, J. Combin.

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Received: September 26, 2006Final Version received: May 16, 2008