Chun-Cheng Chen
Department of Mathematics National Central University
1
Outline
Introduction
Packing and covering of λKn
Packing and covering of λKn,n
2
Introduction
Kn : the complete graph with n vertices.
Km,n : the complete bipartite graph with parts of
sizes m and n.
If m = n, the complete bipartite graph is
referred to as balanced.
Sk (k-star) : the complete bipartite graph K1,k.
Pk (k-path) : a path on k vertices.
Ck (k-cycle) : a cycle of length k.
3
λH : the multigraph obtained from H by replacing
each edge e by λ edges each having the same
endpoints as e.
When λ = 1, 1H is simply written as H.
A decomposition of H : a set of edge-disjoint
subgraphs of H whose union is H.
A G-decomposition of H : a decomposition of H in
which each subgraph is isomorphic to G.
If H has a G-decomposition, we say that H is Gdecomposable and write G|H.
4
Example.
K6
K3,4
P4|K6
S3|K3,4
5
An (F, G)-decomposition of H : a decomposition of
H into copies of F and G using at least one of each.
If H has an (F, G)-decomposition, we say that H is
(F, G)-decomposable and write (F, G)|H.
6
Example.
K6
K3,4
(P4, S3)|K6
(P4, S3)|K3,4
7
Abueida and Daven [3] investigated the problem of
(Kk, Sk)-decomposition of the complete graph Kn.
[3] A. Abueida and M. Daven, Multidecompositons of the complete graph, Ars
Combin. 72(2004), 17-22 .
Abueida and Daven [4] investigated the problem of
the (C4, E2)-decomposition of several graph products
where E2 denotes two vertex disjoint edges.
[4] A. Abueida and M. Daven, Multidecompositions of several graph products,
Graphs Combin. 29 (2013), 315-326.
Abueida and O‘Neil [7] settled the existence problem
for (Ck, Sk-1)-decomposition of the complete
multigraph λKn for k ∈ {3,4,5}.
[7] A. Abueida and T. O'Neil, Multidecomposition of λKm into small cycles and claws,
Bull. Inst. Combin. Appl. 49 (2007), 32-40.
8
Priyadharsini and Muthusamy [15, 16] gave
necessary and sufficient conditions for the existence
of (Gn,Hn)-decompositions of λKn and λKn,n where Gn,Hn
∈ {Cn,Pn,Sn-1}.
[15] H. M. Priyadharsini and A. Muthusamy, (Gm,Hm)-multifactorization of λKm, J.
Com-bin. Math. Combin. Comput. 69 (2009), 145-150.
[16] H. M. Priyadharsini and A. Muthusamy, (Gm,Hm)-multidecomposition of
Km,m(λ),Bull. Inst. Combin. Appl. 66 (2012), 42-48.
9
A graph-pair (G,H) of order m is a pair of nonisomorphic graphs G and H on m non-isolated
vertices such that G∪H is isomorphic to Km.
Abueida and Daven [2] and Abueida, Daven and
Roblee [5] completely determined the values of n for
which λKn admits a (G,H)-decomposition where (G,H)
is a graph-pair of order 4 or 5.
[2] A. Abueida and M. Daven, Multidesigns for graph-pairs of order 4 and 5, Graphs
Com-bin. 19 (2003), 433-447.
[5] A. Abueida, M. Daven, and K. J. Roblee, Multidesigns of the λ-fold complete
graph for graph-pairs of order 4 and 5, Australas J. Combin. 32 (2005), 125-136.
10
Abueida, Clark and Leach [1] and Abueida and
Hampson [6] considered the existence of
decompositions of Kn−F for the graph-pair of order 4
and 5, respectively, where F is a Hamiltonian cycle, a
1-factor, or almost 1-factor.
[1] A. Abueida, S. Clark, and D. Leach, Multidecomposition of the complete graph
into graph pairs of order 4 with various leaves, Ars Combin. 93 (2009), 403-407.
[6] A. Abueida and C. Hampson, Multidecomposition of Kn − F into graph-pairs of
order 5 where F is a Hamilton cycle or an (almost) 1-factor, Ars Combin. 97
(2010), 399-416
Shyu [17] investigated the problem of decomposing
Kn into paths and stars with k edges, giving a
necessary and sufficient condition for k = 3.
[17] T.-W. Shyu, Decomposition of complete graphs into paths and stars, Discrete
Math. 310 (2010), 2164-2169.
11
In [18, 19], Shyu considered the existence of a
decomposition of Kn into paths and cycles with k
edges, giving a necessary and sufficient condition for
k ∈ { 3, 4}.
[18] T.-W. Shyu, Decompositions of complete graphs into paths and cycles, Ars
Combin. 97(2010), 257-270.
[19] T.-W. Shyu, Decomposition of complete graphs into paths of length three and
triangles, Ars Combin. 107 (2012), 209-224.
Shyu [20] investigated the problem of decomposing
Kn into cycles and stars with k edges, settling the case
k = 4.
[20] T.-W. Shyu, Decomposition of complete graphs into cycles and stars, Graphs
Combin. 29 (2013), 301-313.
12
In [21], Shyu considered the existence of a
decomposition of Km,n into paths and stars with k
edges, giving a necessary and sufficient condition for
k = 3.
[21] T.-W. Shyu, Decomposition of complete bipartite graphs into paths and stars
with same number of edges, Discrete Math. 313 (2013), 865-871.
Lee [12] and Lee and Lin [13] established necessary
and sufficient conditions for the existence of (Ck,Sk)decompositions of the complete bipartite graph and
the complete bipartite graph with a 1-factor removed,
respectively.
[12] H.-C. Lee, Multidecompositions of complete bipartite graphs into cycles and
stars, Ars Combin. 108 (2013), 355-364.
[13] H.-C. Lee and J.-J. Lin, Decomposition of the complete bipartite graph with a 1factor removed into cycles and stars, Discrete Math. 313 (2013), 2354-2358.
13
When a multigraph H does not admit an (F,G)decomposition, two natural questions arise :
(1) What is the minimum number of edges needed
to be removed from the edge set of H so that the
resulting graph is (F,G)-decomposable?
(2) What is the minimum number of edges needed
to be added to the edge set of H so that the
resulting graph is (F,G)-decomposable?
These questions are respectively called the maximum
packing problem and the minimum covering problem
of H with F and G.
14
Let F, G, and H be multigraphs. For subgraphs L and R
of H.
An (F,G)-packing ((F,G)-covering) of H with leave L
(padding R) is an (F,G)-decomposition of H−E(L)
(H+E(R)).
An (F,G)-packing ((F,G)-covering) of H with the largest
(smallest) cardinality is a maximum (F,G)-packing
(minimum (F,G)-covering) of H, and its cardinality is the
(F,G)-packing number ((F,G)-covering number) of H,
denoted by p(H;F,G) (c(H;F,G)).
An (F,G)-decomposition of H is a maximum (F,G)-packing
(minimum (F,G)-covering) with leave (padding) the
empty graph.
15
Example.
K6
(P5, S4)|K6 –E(P4)
(P5, S4)|K6 +E(P2)
16
Abueida and Daven [3] obtained the maximum
packing and the minimum covering of the complete
graph Kn with (Kk,Sk).
[3] A. Abueida and M. Daven, Multidecompositons of the complete graph, Ars
Combin. 72(2004), 17-22 .
Abueida and Daven [2] and Abueida, Daven and Roblee
[5] gave the maximum packing and the minimum
covering of Kn and λKn with G and H, respectively,
where (G,H) is a graph-pair of order 4 or 5.
[2] A. Abueida and M. Daven, Multidesigns for graph-pairs of order 4 and 5, Graphs
Com-bin. 19 (2003), 433-447.
[5] A. Abueida, M. Daven, and K. J. Roblee, Multidesigns of the λ-fold complete
graph for graph-pairs of order 4 and 5, Australas J. Combin. 32 (2005), 125-136.
17
Packing and covering of λKn
Let G be a multigraph.
The degree of a vertex x of G (degG(x)) : the number
of edges incident with x.
The center of Sk : the vertex of degree k in Sk.
The endvertex of Sk : the vertex of degree 1 in Sk.
(x;y1,y2,…,yk) : the Sk with center x and endvertices
y1,y2,…,yk.
19
v1v2… vk : the Pk through vertices v1,v2,…,vk in order, and
the vertices v1 and vk are referred to as its origin and
terminus.
If P = x1x2…xt, Q = y1y2…ys and xt = y1, then P + Q =
x1x2…xty2…ys.
Pk(v1,vk) : a Pk with origin v1 and terminus vk.
20
G[U] : the subgraph of G induced by U.
G[U,W] : the maximal bipartite subgraph of G with
bipartition (U,W).
When G1,G2,…,Gt are multigraphs, not necessarily
disjoint.
𝑡
G1∪G2∪…∪Gt or i=1 Gi : the graph with vertex set
𝑡
𝑡
V
(G
i) and edge set
i=1
i=1 E(Gi).
When the edge sets are disjoint, G = 𝑡i=1 Gi expresses
the decomposition of G into G1,G2,…,Gt.
21
Proposition 2.1.
For positive integers λ, n, and t, and any sequence
m1,m2,…,mt of positive integers, the complete
multigraph λKn can be decomposed into paths of
lengths m1,m2,…,mt if and only if mi ≤ n-1 and
m1+m2+…+mt = |E(λKn)|.
[9] Darryn Bryant, Packing paths in complete graphs, J. Combin. Theory Ser. B 100
(2010), 206-215.
22
Proposition 2.2.
For a positive even integer n and V (Kn) = {x , x ,…, x },
𝑛
the complete graph Kn can be decomposed into copies
2
of n-paths Pn(x ,x ), Pn(x ,x ) ,…, Pn(x , x ).
0
0
𝑛
2
1
1+
𝑛
2
𝑛
2
−1
1
n-1
n-1
[8] J. Bosák, Decompositions of Graphs, Kluwer, Dordrecht, Netherlands, 1990.
[10] P. Hell and A. Rosa, Graph decompositions, handcuffed prisoners and balanced
P-designs, Discrete Math. 2 (1972), 229-252.
23
Theorem 2.5.
Let n and k be positive integers and let t be a
nonnegative integer with k ≥ 3,n ≥ k + 2, and t ≤ k-1.
If |E(Kn)| ≡ t (mod k), then Kn has a (Pk+1, Sk)-packing
with leave Pt+1.
Proof.
Let n = qk+r and |E(Kn)| = pk+t with integers q,p,r and 0 ≤ r ≤ k -1.
Now we consider the case r ≠ 1, we set n = m + sk + 1 where m and s
are positive integers with 1 ≤ m ≤ k-1.
Case 1. m is odd.
Case 2. m is even.
25
Proposition 2.1.
Lemma 2.4.
For
positive
n, and integers
t, and any
sequence
Proposition
Let
k, m, p, integers
and2.2.
s be λ,
positive
and
let t be a
Case 2. m is even.
Kn = Km+sk+1 = Km∪Km,sk+1∪Ksk+1.
𝑚
2
By Proposition 2.2, Km= Pm.
mnonnegative
1,m2,…,mt of positive integers, the complete
with max{m,t}
For a positive integer
even integer
n and V (K≤nk-1.
) = {x0,x1,…,xn-1},
multigraph λK𝑚(𝑚−1)
n can be decomposed
into
paths of𝑛
𝑠𝑘+1 𝑠𝑘
the
complete
graph K+n can
be decomposed
into
copies
If
𝑝𝑘
+
𝑡
=
+
𝑚(𝑠𝑘
+
1)
then
2
lengths m1,m2,…,m
2 t if and only
2 if mi ≤ n-1 and m1+
of
n-paths |E(λK
Pn0.(x0,xn)|.
), Pn(x1,x1+ ) ,…, Pn(x −1,x ).
p-(s+1)m
m
2+…+mt = >
𝑛
𝑛
𝑛
2
2
2
𝑚
2
n-1
𝑚
2
By Proposition 2.1 and Lemma 2.4, Ksk+1= Pk-m ∪(p-ms- )Pk+1∪Pt+1.
P𝑘 − 𝑚
…
Ksk+1
Sk|K1,sk
Pk+1
Km
Sk|K1,sk
…
…
P𝑚
Hence, Kn = msSk∪(p-ms)Pk+1∪Pt+1.
30
Theorem 2.6.
Let λ, n and k be positive integers and let t be a
nonnegative integer with k ≥ 3, n ≥ k+2, and t ≤ k-1. If
|E(λKn)| ≡ t (mod k), then λKn has a (Pk+1, Sk)-packing
with leave Pt+1.
31
Corollary 2.7.
For positive integers λ, k, and n with k ≥ 3, the
complete multigraph λKn is (Pk+1,Sk)-decomposable if
𝜆𝑛(𝑛−1)
and only if n ≥ k + 1,
≡ 0 (mod k) and (λ,n) ≠
2
(1, k + 1).
33
For positive integers k, n, and t with k ≤ n-1 and t ≤ k-1.
If λKn has a (Pk+1, Sk)-packing with leave Pt+1, then it has a (Pk+1, Sk)covering with padding Pk-t+1.
Since p(λKn; Pk+1,Sk) ≤
𝜆𝑛(𝑛−1)
2𝑘
following by Theorem 2.6.
≤
𝜆𝑛(𝑛−1)
2𝑘
≤ c(λKn; Pk+1,Sk), we have the
Theorem 2.8.
If λ, n, and k are positive integers with n ≥ k+2 ≥ 5,
𝜆𝑛(𝑛−1)
then p(λKn; Pk+1,Sk) =
and c(λKn; Pk+1, Sk)
=
𝜆𝑛(𝑛−1)
2𝑘
2𝑘
.
38
Packing and covering of λKn,n
(v1,v2,…,vk) : the k-cycle of G through vertices
v1,v2,…,vk in order.
μ(uv) : the number of edges of G joining u and v.
A central function c from V(G) to the set of nonnegative integers is defined as follows : for each v∈
V(G), c(v) is the number of Sk in the decomposition
whose center is v.
39
Proposition 3.1.
For a positive integer k, a multigraph H has an Skdecomposition with central function c if and only if let
c : V → N ∪ {0}.
(i) 𝑘 𝑣∈𝑉(𝐻) 𝑐 𝑣 = |𝐸(𝐻)|;
(ii) for all 𝑥, 𝑦 ∈ 𝑣 𝐻 , 𝜇 𝑥𝑦 ≤ 𝑐 𝑥 + 𝑐(𝑦);
(iii) for all 𝑆⊆V(H),
𝑘 𝑣∈𝑆 𝑐 𝑣 ≤ 𝜀 𝑆 + 𝑥𝜖𝑆,𝑦𝜖𝑉 𝐻 \S min{𝑐 𝑥 , 𝜇(𝑥𝑦)} .
where 𝜀 𝑆 denotes the number of edges of H with both
ends in S.
[11] D.G. Hoffman, The real truth about star designs, Discrete Math. 284 (2004),
177-180.
40
Lemma 3.4.
Let λ, k, and n be positive integers with 3 ≤ k < n < 2k.
If 𝜆(𝑛 − 𝑘)2 < 𝑘 , then λKn,n has a (Pk+1, Sk)-packing P
with |P|=
𝜆𝑛2
𝑘
and a (Pk+1, Sk)-covering C with |C|=
𝜆𝑛2
𝑘
.
Lemma 3.6.
Let λ, k, and n be positive integers with 3 ≤ k < n < 2k.
If 𝜆(𝑛 − 𝑘)2 ≥ 𝑘 , then λKn,n has a (Pk+1, Sk)-packing P
with |P|=
𝜆𝑛2
𝑘
and a (Pk+1, Sk)-covering C with |C|=
𝜆𝑛2
𝑘
44
.
Proof of Lemma 3.4.
Let n = k + r. The assumption k < n < 2k implies 0 < r < k.
Packing.
≅ λ𝑟𝑆𝑘
Lemma 3.3.
λKn,n = λKk,k ∪ λKk,r ∪ λKr,k ∪ λKr,r
By Lemma 3.3.⇒ (Pk+1, Sk) |Kk,k
leave
If λ and k are positive integers
with k ≥ 2, then λKk,k is (Pk+1, Sk)decomposable.
𝜆𝑟 2 = 𝜆(𝑛 − 𝑘)2 < 𝑘
45
Covering
=𝜆𝑟 2 .
Let s
Claim. λKn,n+E(Pk-s+1)=Pk+1∪
𝜆𝑛2
𝑘
Sk.
Let 𝐴0 = 𝑎0 , 𝑎1 , … , 𝑎 𝑠−1 , 𝐴1 = 𝐴 − 𝐴0 , 𝐵0 = 𝑏0 , 𝑏1 , … , 𝑏𝑘−1 and
2
𝐵1 = 𝐵 − 𝐵0
Define a (k + 1)-path P as follows:
P=
𝑏0 𝑎0 𝑏1 𝑎1 … 𝑏 𝑘−1 𝑎 𝑘−1
𝑖𝑓 𝑘 𝑖𝑠 𝑜𝑑𝑑,
𝑏0 𝑎0 𝑏1 𝑎1 … 𝑏 𝑘−1 𝑎 𝑘−1 𝑏 𝑘−1
𝑖𝑓 𝑘 𝑖𝑠 𝑒𝑣𝑒𝑛
2
2
2
2
2
Let P’ be the (k-s+1)-subpath of P with origin 𝑎 𝑘−1 for odd k and with
2
origin 𝑏 𝑘−1 for even k.
2
𝑎0
𝐴0
P
P’
𝐵0
𝑏0
𝑎1
𝑏1
𝑎2
𝑏2
…
…
…
…
s is odd.
k is odd.
H = λKn,n - E(P) + E(P’).
𝑎 𝑠−1
2
𝑏 𝑠−1
2
−1
−1
𝑎 𝑠−1 𝑎 𝑠−1
2
2
𝑏 𝑠−1 𝑏 𝑠−1
2
2
… 𝑎 𝑘−1 −1 𝑎 𝑘−1
2
… 2
…
… 𝑏 𝑘−1 −1 𝑏 𝑘−1
2
2
𝑎 𝑘−1
2
𝑏 𝑘−1
2
…
…
…
…
𝑎𝑘−1
𝑏𝑘−1
…
…
…
…
𝑎𝑛−1
𝐴1
𝐵1
𝑏𝑛−1
46
Claim. Sk|H
Proposition 3.1.
For a positive integer k, a multigraph H has an Sk-decomposition with
central function c if and only if let c : V → N∪{0}.
(i) 𝑘 𝑣∈𝑉(𝐻) 𝑐 𝑣 = |𝐸(𝐻)|;
(ii) for all 𝑥, 𝑦 ∈ 𝑣 𝐻 , 𝜇 𝑥𝑦 ≤ 𝑐 𝑥 + 𝑐(𝑦);
(iii) for all 𝑆⊆V(H), 𝑘 𝑣∈𝑆 𝑐 𝑣 ≤ 𝜀 𝑆 + 𝑥𝜖𝑆,𝑦𝜖𝑉 𝐻 \S min{𝑐 𝑥 , 𝜇(𝑥𝑦)}.
where 𝜀 𝑆 denotes the number of edges of H with both ends in S.
Define a function c : V (H)→N∪{0} as follows :
𝑐 𝑣 =
0
λ
𝑖𝑓 𝑣 ∈ 𝐵0 ,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
47
Theorem 3.7.
If λ, k, and n are positive integers with 3 ≤ k ≤ n, then
p(λKn,n; Pk+1, Sk) =
𝜆𝑛2
𝑘
and c(λKn,n; Pk+1, Sk ) =
𝜆𝑛2
𝑘
.
Corollary 3.8.
For positive integers λ, k and n with k≥3, the balanced
complete bipartite multigraph λKn,n is (Pk+1, Sk)decomposable if and only if k ≤ n and 𝜆𝑛2 is divisible
by k.
48
[1] A. Abueida, S. Clark, and D. Leach, Multidecomposition of the complete graph
into graph pairs of order 4 with various leaves, Ars Combin. 93 (2009), 403-407.
[2] A. Abueida and M. Daven, Multidesigns for graph-pairs of order 4 and 5, Graphs
Com-bin. 19 (2003), 433-447.
[3] A. Abueida and M. Daven, Multidecompositons of the complete graph, Ars
Combin. 72(2004), 17-22 .
[4] A. Abueida and M. Daven, Multidecompositions of several graph products,
Graphs Combin. 29 (2013), 315-326.
[5] A. Abueida, M. Daven, and K. J. Roblee, Multidesigns of the λ-fold complete
graph for graph-pairs of order 4 and 5, Australas J. Combin. 32 (2005), 125-136.
[6] A. Abueida and C. Hampson, Multidecomposition of Kn − F into graph-pairs of
order 5 where F is a Hamilton cycle or an (almost) 1-factor, Ars Combin. 97 (2010),
399-416
[7] A. Abueida and T. O'Neil, Multidecomposition of λKm into small cycles and claws,
Bull. Inst. Combin. Appl. 49 (2007), 32-40.
[8] J. Bosák, Decompositions of Graphs, Kluwer, Dordrecht, Netherlands, 1990.
[9] Darryn Bryant, Packing paths in complete graphs, J. Combin. Theory Ser. B 100
(2010), 206-215.
[10] P. Hell and A. Rosa, Graph decompositions, handcuffed prisoners and balanced
P-designs, Discrete Math. 2 (1972), 229-252.
[11] D.G. Hoffman, The real truth about star designs, Discrete Math. 284 (2004),
177-180.
[12] H.-C. Lee, Multidecompositions of complete bipartite graphs into cycles and
stars, Ars Combin. 108 (2013), 355-364.
[13] H.-C. Lee and J.-J. Lin, Decomposition of the complete bipartite graph with a 1factor removed into cycles and stars, Discrete Math. 313 (2013), 2354-2358.
[14] C. A. Parker, Complete bipartite graph path decompositions, Ph.D. Thesis,
Auburn Uni-versity, Auburn, Alabama, 1998.
[15] H. M. Priyadharsini and A. Muthusamy, (Gm,Hm)-multifactorization of λKm, J.
Com-bin. Math. Combin. Comput. 69 (2009), 145-150.
[16] H. M. Priyadharsini and A. Muthusamy, (Gm,Hm)-multidecomposition of
Km,m(λ),Bull. Inst. Combin. Appl. 66 (2012), 42-48.
[17] T.-W. Shyu, Decomposition of complete graphs into paths and stars, Discrete
Math. 310 (2010), 2164-2169.
[18] T.-W. Shyu, Decompositions of complete graphs into paths and cycles, Ars
Combin. 97(2010), 257-270.
[19] T.-W. Shyu, Decomposition of complete graphs into paths of length three and
triangles, Ars Combin. 107 (2012), 209-224.
[20] T.-W. Shyu, Decomposition of complete graphs into cycles and stars, Graphs
Combin. 29 (2013), 301-313.
[21] T.-W. Shyu, Decomposition of complete bipartite graphs into paths and stars
with same number of edges, Discrete Math. 313 (2013), 865-871.
Thank you for your listening.
52