AbstractGiven a graph G, a G-decomposition of the complete graph Kv is a set of graphs, all isomorphic to G, whose edge sets partition the edge set of Kv. A G-decomposition of Kv is also called a G-design and the graphs of the partition are said to be the blocks. A G-design is said to be balanced if the number of blocks containing any given vertex of Kv is a constant.In this paper the concept of strongly balanced G-design is introduced and strongly balanced path-designs are studied. Furthermore, we determine the spectrum of those path-designs which are balanced, but not strongly balanced

Berardi, Luigia

Gionfriddo, Mario

Rota, Rosaria

Elsevier - Publisher Connector 

Discrete Mathematics 312 (2012) 633–636
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Discrete Mathematics
journal homepage: www.elsevier.com/locate/disc
Balanced and strongly balanced Pk-designs✩
Luigia Berardi a, Mario Gionfriddo b, Rosaria Rota c
a Dipartimento di Ingegneria Elettrica e dell’Informazione, Universitá dell’Aquila, Italy
b Dipartimento di Matematica e Informatica, Universitá di Catania, Italy
c Dipartimento di Matematica, Universitá di RomaTre, Italy
a r t i c l e i n f o
Article history:
Available online 30 June 2011
Keywords:
Balanced
Strongly balanced
Simply balanced
G-design
a b s t r a c t
Given a graph G, a G-decomposition of the complete graph Kv is a set of graphs, all
isomorphic to G, whose edge sets partition the edge set of Kv . A G-decomposition of Kv
is also called a G-design and the graphs of the partition are said to be the blocks. A G-design
is said to be balanced if the number of blocks containing any given vertex of Kv is a constant.
In this paper the concept of strongly balanced G-design is introduced and strongly
balanced path-designs are studied. Furthermore, we determine the spectrum of those path-
designs which are balanced, but not strongly balanced.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
Let Kv = (X, E) be the complete graph defined on a vertex set X . Let G be a subgraph of Kv . A G-decomposition of Kv is a
pairΣ = (X,B), whereB is a partition of the edge set of Kv into subsets all of which yield subgraphs that are isomorphic
to G. A G-decompositionΣ = (X,B) of Kv is also called a G-design of order v and the classes of the partitionB are said to
be the blocks ofΣ [4–6].
A G-design is called balanced if the degree of each vertex x ∈ X is a constant: in other words, the number of blocks of Σ
containing x is a constant [1,3].
In the caseG = Pk, pathwith k vertices, aG-decomposition is called a path-decomposition or a Pk-decomposition or a Pk-design.
In this paperwe introduce the concept of a strongly balanced G-design as a balancedG-designwith an additional property.We
study some properties and relations between balanced and strongly balanced G-designs and, also, determine the spectrum
of those path-designs which are balanced, but not strongly balanced.
2. Main definitions
In what follows, we denote by Pk a path with k vertices. If a Pk has vertices x1, x2, . . . , xk and edges {x1, x2},
{x2, x3}, . . . , {xk−1, xk}, it will be denoted by (x1, . . . , xk).
It is well known that a Pk-design of order v, briefly a Pk(v)-design, exists if and only if [2]: v · (v−1) ≡ 0, mod 2 · (k−1),
v ≥ k.
Let G be a graph with n vertices and let Σ = (X,B) be a G-design. If x ∈ X is a vertex of Σ , the degree d(x) of x is the
number of blocks ofΣ containing x.
A G-designΣ = (X,B) is said to be balanced if the degree d(x) of a vertex x ∈ X is a constant.
Observe that if G is a regular graph then a G-design is always balanced, hence the notion of a balanced G-design becomes
meaningful only for a non-regular graph G.
✩ Lavoro eseguito nell’ambito del GNSAGA e nell’ambito del MIUR.
E-mail addresses: luigia.berardi@ing.univaq.it (L. Berardi), gionfriddo@dmi.unict.it (M. Gionfriddo), rota@mat.uniroma3.it (R. Rota).
0012-365X/$ – see front matter© 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.disc.2011.05.015
634 L. Berardi et al. / Discrete Mathematics 312 (2012) 633–636
Example 1. LetΣ = (X,B) be the P6(6)-design defined on Z6, having the blocks:
B1 = (0, 1, 5, 2, 4, 3), B2 = (1, 2, 3, 5, 0, 4), B3 = (2, 0, 3, 1, 4, 5).
Σ is a balanced P6-design of order 6. 
Example 2. LetΣ = (X,B) be the P4(7)-design defined on X = Z7, having the blocks:
B = (i, i+ 1, i+ 3, i+ 6),
for every i ∈ Z7.
Σ is a balanced P4-design of order 7. 
Example 3. LetΣ = (X,B) be the P3(4)-design defined on X = {0, 1, 2, 3}, having the blocks:
B1 = (0, 1, 2), B2 = (0, 2, 3), B3 = (0, 3, 1).
Σ is not a balanced P3-design. The vertex 0 has degree three and the vertex 1 has degree two. 
3. Strongly balanced and simply balanced G-designs
Let G be a graph and let A1, A2, . . . , Ah be the orbits of the automorphism group of G on its vertex set. LetΣ = (X,B) be
a G-design.
We define the degree dAi(x) of a vertex x ∈ X as the number of blocks ofΣ containing x as an element of Ai.
We say that:
Σ = (X,B) is a strongly balanced G-design if, for every i = 1, 2, . . . , h, there exists a constant Ci such that
dAi(x) = Ci,
for every x ∈ X .
Clearly, since for each vertex x ∈ X the relation d(x) =hi=1 dAi(x) holds, we have:
Theorem 3.1. A strongly balanced G-design is a balanced G-design.
The P6(6)-design of Example 1 is balanced but not strongly balanced, showing that the converse of Theorem 3.1 does not
hold.
We say that a G-design is simply balanced if it is balanced, but not strongly balanced.
Example 4. Given a kite K3+e having vertices a, b, c, d, we denote it by [a, b, (c, d)], where the ordered pair (c, d) contains
the vertex c of degree three and the vertex d of degree one.
LetΣ = (X,B) be the (K3 + e)-design of order 9, defined on X = Z9, having the blocks:
[0, 2, (1, 8)], [0, 3, (6, 1)], [0, 4, (7, 2)],
[0, 5, (8, 4)], [4, 5, (1, 7)], [6, 8, (2, 5)],
[7, 8, (3, 1)], [2, 3, (4, 6)], [6, 7, (5, 3)].
Σ is a balanced (K3 + e)-design, but it is not strongly balanced: observe that the vertex 0 is never a terminal vertex, while
the vertex 1 occurs twice in a terminal position. 
In what follows, we will determine the spectrum for simply balanced P5-designs and P6-designs. Our purpose is a
preliminary study of the existence of balanced paths-designs which are not strongly balanced. The same investigation for
(K3 + e)-designs and other designs will be the subject of future work.
4. Simply balanced P5-designs
It is known that [2].
Theorem 4.1. A balanced P5(v)-design exists if and only if v ≡ 1 mod 8, v ≥ 9.
Proof. The proof can be obtained using the difference method. If v = 8k+ 1, k ≥ 1, we can consider a P5(v)-design, defined
on Z8k+1, having as blocks the translates of k base blocks. These can be obtained by distributing in a suitable way the 4k
differences among different elements of Z8k+1 on the 4k edges of the k base blocks. 
By the construction indicated in the previous Theorem we obtain P5(v)-designs which are strongly balanced, because
the difference method produces these types of designs.
L. Berardi et al. / Discrete Mathematics 312 (2012) 633–636 635
So, we can say that:
Theorem 4.2. A strongly balanced P5(v)-design exists if and only if v ≡ 1 mod 8, v ≥ 9.
So, we must determine the spectrum of those P5(v)-designs which are balanced, but not strongly balanced.
Theorem 4.3. There exists a simply balanced P5(9)-design.
Proof. Let v = 9. The systemΣ = (Z9,B) having the following blocks:
(0, 1, 3, 7, 4), (1, 2, 4, 8, 5), (2, 3, 0, 5, 6),
(3, 4, 6, 1, 7), (4, 5, 7, 2, 8), (0, 6, 8, 3, 5),
(6, 7, 0, 4, 1), (7, 8, 1, 5, 2), (8, 0, 2, 6, 3),
is a balanced P5(9)-design, it is not strongly balanced. 
Theorem 4.4. A simply balanced P5(v)-design exists if and only if v ≡ 1 mod 8, v ≥ 9.
Proof (Construction). v → v + 8
Let Σ1 = (X,B1) be a simply balanced P5(9)-design. Let Σ2 = (Y ,B2) be a strongly balanced P5(v)-design, where
v = 8k + 1, k ≥ 1 and X ∩ Y = {0}. Define a family F of P5-paths, containing all the pairs {x, y} with x ∈ X − {0} and
y ∈ Y − {0}, as follows.
Since v = 8k + 1, then |Y − {0}| = 8k and it is possible to partition the set Y − {0} into two classes as follows:
Γa = {a1, a2, . . . , a4k}, Γb = {b1, b2, . . . , b4k}.
Furthermore, since |X − {0}| = 8, it is possible to partition X − {0} into the following two classes: A = {α1, α2, α3, α4},
B = {β1, β2, β3, β4}.
Let Γα,β be the family containing the following blocks:
(α4, ai, α1, ak+i, α3),
(α4, ak+i, α2, ai, α3),
(α1, a2k+i, α3, a3k+i, α2),
(α1, a3k+i, α4, a2k+i, α2),
(β4, bi, β1, bk+i, β3),
(β4, bk+i, β2, bi, β3),
(β1, b2k+i, β3, b3k+i, β2),
(β1, b3k+i, β4, b2k+i, β2),
for every i = 1, 2, . . . , k.
Let Γa,b be the family containing the following blocks:
(b1, α1, b2k+1, α2, b2),
(b2, α1, b2k+2, α2, b3),
. . .
(b2k−1, α1, b4k−1, α2, b2k),
(b2k, α1, b4k, α2, b1),
(b2k+1, α3, b1, α4, b2k+2),
(b2k+2, α3, b2, α4, b2k+3),
. . .
(b4k−1, α3, b2k−1, α4, b4k),
(b4k, α3, b2k, α4, b2k+1),
(a1, β1, a2k+1, β2, a2),
(a2, β1, a2k+2, β2, a3),
. . .
(a2k−1, β1, a4k−1, β2, a2k),
(a2k, β1, a4k, β2, a1),
636 L. Berardi et al. / Discrete Mathematics 312 (2012) 633–636
(a2k+1, β3, a1, β4, a2k+2),
(a2k+2, β3, a2, β4, a2k+3),
. . .
(a4k−1, β3, a2k−1, β4, a4k),
(a4k, β3, a2k, β4, a2k+1).
IfF = Γα,β ∪Γa,b, it is possible to verify that the blocks ofF cover all the edges having a vertex in X −{0} and the other
vertex in Y − {0}. Furthermore, each vertex x ∈ X − {0} appears in F k times in the central position, 2k times in any of the
two medial positions and 2k times in any of the two extreme positions of a P5-block.
SetB = B1 ∪B2 ∪F and note that the blocks ofB passing through an element of X − {0} are only those inB1 ∪F . Since
Σ1 is simply balanced one can easily check thatΣ = (X ∪ Y ,B) is a simply balanced P5(v + 8)-design. 
5. Simply balanced P6-designs
It is known that [2].
Theorem 5.1. A balanced P6(v)-design exists if and only if v ≡ 1 mod 5, v ≥ 6.
Theorem 5.2. There exist strongly balanced and simply balanced P6(6)-designs.
Proof. Let v = 6. We can verify that:
(i) the systemΣ1 = (Z6,B1)with the blocks:
(0, 1, 2, 5, 4, 3), (1, 5, 3, 0, 2, 4), (2, 3, 1, 4, 0, 5),
is a strongly balanced P6(9)-design;
(ii) the systemΣ2 = (Z6,B2)with the blocks:
(0, 1, 2, 4, 5, 3), (1, 5, 2, 0, 3, 4), (2, 3, 1, 4, 0, 5),
is a balanced P6(6)-design, which is not strongly balanced. 
Theorem 5.3. (i) A simply balanced P6(v)-design exists if and only if v ≡ 1 mod 5, v ≥ 6. (ii) A strongly balanced P6(v)-design
exists if and only if v ≡ 1 mod 5, v ≥ 6.
Proof. From Theorem 5.2, for v = 6 there exist simply balanced P6(6)-designs and strongly balanced P6(6)-designs.
Construction v → v + 5
Observe that if Σ = (X,B) is a strongly balanced P6(v)-design with v = 5k + 1, necessarily all vertices have degree k
in any position of P6. This is not true in any simply balanced P6(v)-design.
Let Σ1 = (X,B1) be a simply [resp. strongly] balanced P6(6)-design. Let Σ2 = (Y ,B2) be a strongly balanced P6(v)-
design, where v = 5k+ 1, k ≥ 1 and X ∩ Y = {0}.
Define a family F of P6-paths, containing all the pairs {x, y}with x ∈ X − {0} and y ∈ Y − {0}, as follows.
Since v = 5k+ 1, |Y − {0}| = 5k and it is possible to partition the set Y − {0} into k classes as follows:
Γi = {ai1, ai2, . . . , ai5},
for every i = 1, 2, . . . , k.
So, if X = {0, A, B, C,D, E}, let F be the family containing the following blocks:
(A, ai1, B, ai5, C, ai4), (B, ai2, C, ai1,D, ai5), (C, ai3,D, ai2, E, ai1),
(D, ai4, E, ai3, A, ai2), (E, ai5, A, ai4, B, ai3),
for every i = 1, 2, . . . , k.
SinceΣ1 is a simply [resp. strongly] balanced P6(6)-design andΣ2 is a strongly balanced P6(v)-design inwhich all vertices
have degree k in any position of P6, if B = B1 ∪ B2 ∪ F , it is possible to verify that Σ = (X ∪ Y ,B) is a simply balanced
P6(v + 9)-design [resp. strongly balanced P6(v + 9)-design with all vertices having degree k+ 1 in any position of P6]. 
References
[1] M. Gionfriddo, G. Quattrocchi, Embedding balanced P3-designs into balanced P4-designs, Discrete Mathematics 308 (2008) 155–160.
[2] K. Heinrich, Path-decompositions, Le Matematiche XLVII (1992) 241–258.
[3] P. Hell, A. Rosa, Graph decompositions, handcuffed prisoners and balanced P-designs, Discrete Mathematics 2 (1972) 229–252.
[4] C. Lindner, Graph decompositions and quasigroups identities, Le Matematiche XLV (1990) 83–118.
[5] C. Rodger, Graph decompositions, Le Matematiche XLV (1990) 119–139.
[6] M. Tarsi, Decompositions of a complete multigraph into simple paths: nonbalanced handcuffed designs, Journal of Combinatorial Theory Ser. A 34
(1983) 60–70.


Balanced and strongly balanced Pk-designs 

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Balanced and strongly balanced Pk-designs

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