On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
On the Balance Index Sets of Permutation
Graphs
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai Conference on Combinatorics
at
Shanghai Jiao Tong University
May 27, 2008
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
the Balance
Conference
Index Sets
on Combinatorics
of PermutationatGraphs
Shanghai Jiao Ton
Abstract
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
Let G be a graph with vertex set V (G ) and edge set E (G ),
and let A = {0, 1}. A labeling f : V (G ) → A induces an edge
partial labeling f ∗ : E (G ) → A defined by f ∗ (xy ) = f (x) if and
only if f (x) = f (y ) for each edge xy ∈ E (G ). We call f is a
friendly labeling if |f −1 (0) − f −1 (1)| = 1. The balance index
set of G , denoted BI(G ), is defined as
{|ef (0) − ef (1)| : |vf (0) − vf (1)| ≤ 1}. In this paper, we study
the balance index sets of the permutation graphs.
Keywords and phrases: vertex labeling, friendly labeling,
cordiality, balance index set, arithmetic progression.
AMS 2000 MSC: 05C78, 05C25
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
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Shanghai Jiao Ton
Vertex Labeling
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
A vertex labeling of a graph G = (V , E ) is a mapping f from
V into the set {0, 1}. For each vertex labeling f of G , we can
define a partial edge labeling f ∗ of G in the following way. For
each edge uv in E , define
(
0 if f (u) = f (v ) = 0,
∗
f (u, v ) =
1 if f (u) = f (v ) = 1.
Note that if f (u) 6= f (v ), then the edge uv is not labeled by
f ∗ . We shall refer f ∗ the induced partial function of f .
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
the Balance
Conference
Index Sets
on Combinatorics
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Shanghai Jiao Ton
Example
On the
Balance Index
Sets of
Permutation
Graphs
The friendly labelings of a graph G with BI(G ) = {0, 1, 2}.
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
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University
1i
0i
0i
@
@
@i
i
1
1
@
@0
@i
1i
0
@
@
1
1 @
1i
@
@
@i
0
|e(0) − e(1)| = 0
Sin-Min Lee and Hsin-hao Su
1i
|e(0) − e(1)| = 1
1i
1
1i
@
@
@i
i
0
0
@
@
1 @
1i
|e(0) − e(1)| = 2
6th
OnShanghai
the Balance
Conference
Index Sets
on Combinatorics
of PermutationatGraphs
Shanghai Jiao Ton
Notations
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
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University
Let vf (0) and vf (1) denote the number of vertices of G that
are labeled 0 and 1 respectively under the mapping f . Similarly,
denoted by ef (0) and ef (1) respectively, the numbers of edges
of G that are labeled 0 and 1 respectively under the induced
partial function f ∗ . In other words, for i = 0, 1,
vf (i) = |{u ∈ V (G ) : f (u) = i}|,
ef (i) = |{uv ∈ E (G ) : f ∗ (uv ) = i}|.
For brevity, when the context is clear, we will simply write v (0),
v (1), e(0) and e(1) without any subscript. We are now ready
to introduce the notion of a balanced graph.
Sin-Min Lee and Hsin-hao Su
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OnShanghai
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Shanghai Jiao Ton
Friendly Labeling
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
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at
Shanghai Jiao
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Definition 1.1. A vertex labeling f of a graph G is said to be
friendly if |vf (0) − vf (1)| ≤ 1, and balanced if both
|vf (0) − vf (1)| ≤ 1 and |ef (0) − ef (1)| ≤ 1.
It is clear that not all the friendly graphs are balanced.
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
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Balanced Index Set
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
Lee, Lee and Ng [7] introduced the following notion in [4] as an
extension of their study of the balanced graphs.
Definition 1.2. The balance index set of the graph G is
defined as
BI(G ) = {|ef (0) − ef (1)| : the vertex labeling f is friendly}.
Sin-Min Lee and Hsin-hao Su
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OnShanghai
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Shanghai Jiao Ton
Example: C4 (3)
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
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For a cycle Cn with vertex set {x1 , x2 , . . . , xn }, we denote by
Cn (t) the cycle with a chord x1 xt .
The balance index sets of C4 (3), C6 (4) and C6 (5) are all equal
to {0, 1}.
x3
1i
1
x2
1i
x3
@
@
@
i 0 @ 0i
x4 0
x1
|e(0) − e(1)| = 0
Sin-Min Lee and Hsin-hao Su
x4
1i
x2
0i
@
@ 1
@
@i
i
0
1
x1
|e(0) − e(1)| = 1
6th
OnShanghai
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Example: C6 (4) and C6 (5)
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
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at
Shanghai Jiao
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University
x3 1i
Z
Z i
i
0 x2
x4 0 Z
Z
0 Z
Zi
i
1 x1
1
x5 0 Z
xZ i
x3 1i
Z
Z i
i
0 x2
x4 0 Z
Z0
Z 0
Z i
i
0 x1
1
x5 1 Z
xZ i
|e(0) − e(1)| = 0
|e(0) − e(1)| = 1
6
x4
1
x3 1h
Z
h
1Z 1h
0
x2
h
h
0 x1
x5 1 Z
Z
h
0
x6 0
|e(0) − e(1)| = 0
Sin-Min Lee and Hsin-hao Su
6
1
x3 0h
Z
Zh
h
0 0
1
x2
1
h 1 1h
x5 1 Z
x1
xZ
6 0h
|e(0) − e(1)| = 1
x4
6th
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Example: Φ(1, 3, 1, 1)
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
The graph Φ(1, 3, 1, 1) is composed of C4 (3) with a pendant
edge appended to each of x1 , x3 and x4 , and three pendant
edges appended to x2 .
The BI(Φ(1, 3, 1, 1)) = {0, 1, 2, 3, 4, 6}.
Note that 5 is missing from the balance index set.
6th Shanghai
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1i
1
1i
1i 0i
0i
0
@
@ 0
1 0i 0i
0
@
@i
1
@
@i
0
1
1i
|e(0) − e(1)| = 0
Sin-Min Lee and Hsin-hao Su
1i
1
1i
1
0i
1i 0i
@
@i i
0
1
@
@
@
@
0 i
i
0
0
1i
|e(0) − e(1)| = 1
6th
OnShanghai
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Example: Φ(1, 3, 1, 1)
On the
Balance Index
Sets of
Permutation
Graphs
1
6th Shanghai
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0i
0
@
@ 0
1 0i 0i
0
1
1i 1i
Sin-Min Lee
and
Hsin-hao Su
1i
1i
1
1i
@
@i
0
@
1@ i
1
|e(0) − e(1)| = 2
1i
1
1i
1i 0i
1
0i
@
@i i
0
1
@
@i
@ i1 @
1
1
|e(0) − e(1)| = 4
Sin-Min Lee and Hsin-hao Su
0i
1
1i 1i
@
@i i
0
1
@
@
@
@
1 i1
i
0
0
1
|e(0) − e(1)| = 3
0i
0i
0i
1
0i
1i
1
1i
0i 1i
1
0i
@
@i i
0
1
@
@
@
@
1 i1
i
1
1
|e(0) − e(1)| = 6
0
0i
6th
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Example: BI(P3 ×L Φ)
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
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at
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The balance index sets depend on the topological structure of
the graphs. We show in the following two graph of the same
order but with different balance index sets.
Let the vertices on P3 be u1 , u2 and u3 , and denoted by St(m),
the star with center c and m pendant vertices. We find that
BI(P3 ×L Φ) = {1, 2, 4} if Φ(u1 ) = Φ(u2 ) = (St(2), c), and
Φ(u3 ) = (St(3), c); but BI(P3 ×L Φ) = {0, 2, 4} if
Φ(u1 ) = Φ(u3 ) = (St(2), c), and Φ(u2 ) = (St(3), c).
Sin-Min Lee and Hsin-hao Su
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Example: Φ(u1 ) = Φ(u2 ) = (St(2), c), and
Φ(u3 ) = (St(3), c)
On the
Balance Index
Sets of
Permutation
Graphs
u1,1
0i
J
Ji
1
Sin-Min Lee
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u1,2 u2,1
0i 0i
u1
u1,1
0i
0
Ji
0
u1
u1,1
0i
0J
0
Ji
0
u2
u1,2 u2,1
0i 0i
0J
u2,2 u3,1 u3,2 u3,3
0i 1i 1i 1i
1J 1
1
Ji
1
u3
u2,2 u3,1 u3,2 u3,3
0i 1i 1i 1i
J
1J 1 1
Ji
Ji
1
1
1
u1,2 u2,1
0i 0i
u2
Sin-Min Lee and Hsin-hao Su
|e(0) − e(1)| = 2
u3
u2,2 u3,1 u3,2 u3,3
0i 0i 1i 1i
J
J
J 1 1
Ji
Ji
Ji
1
1
1
1
1
u1
|e(0) − e(1)| = 1
u2
|e(0) − e(1)| = 4
u3
6th
OnShanghai
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Example: Φ(u1 ) = Φ(u3 ) = (St(2), c), and
Φ(u2 ) = (St(3), c)
On the
Balance Index
Sets of
Permutation
Graphs
u1,1
0i
u1,2 u2,1 u2,2 u2,3 u3,1
u3,2
0i 0i 0i 1i 1i
1i
J
|e(0) − e(1)| = 0
1
0J 0 1J
Ji
Ji
Ji
1u
0u
1u
Sin-Min Lee
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6th Shanghai
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1
2
3
u1,2 u2,1 u2,2 u2,3 u3,1
u3,2
0i 0i 0i 1i 1i
1i
0
J
1
J
1
1 |e(0) − e(1)| = 2
Ji
Ji
J
1
0u
1u
1iu
u1,1
0i
0J
1
2
3
u1,1
0i
u1,2 u2,1 u2,2 u2,3 u3,1
u3,2
0i 0i 0i 0i 1i
1i
J
J
|e(0) − e(1)| = 4
1
1J
Ji
Ji
Ji
1
1
1u
1u
1u
1
Sin-Min Lee and Hsin-hao Su
2
3
6th
OnShanghai
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Permutation Graphs
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
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Let π be a permutation of the set [n] = {1, 2, . . . , n}. We
denote π(i) to be the image of i. For each graph G of order n
and a permutation π of the set [n]. The π-permutation graph
of G is the graph union of two disjoint copies of G , namely GT
and GB , together with the edges joining vertex vi of GT with
vπ(i) of GB (See [2], p.175).
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Generalized Permutation Graphs
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
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Let G and H be two graphs with the same number of vertices.
Let π be a permutation between the vertices of G and H. The
edges between two graphs are called permutation edges.
Those are the edges which have one vertex v in G and another
vertex π(v ) in H.
The generalized permutation graph of G and H is defined
by the disjoin union of the G and H with their permutation
edges, and denoted by Perm(G , π, H).
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Example: Perm(K4 , id)
On the
Balance Index
Sets of
Permutation
Graphs
The BI(Perm(K4 , id)) = {0}.
Sin-Min Lee
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u3
1i
u2 0i 1
v2
0i 0
A 1
A
0 A u4
i
1
i
u1 0
v3
1i
1
0
A 1
A
0 A v4
i
1
i
0 v1
|e(0) − e(1)| = 0
Sin-Min Lee and Hsin-hao Su
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Lemma
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
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Lemma
S
Let f be a friendly labeling of the disjoint union G H of two
graphs G and H, where G and H have the same number of
vertices. Then, the number of 0-vertex of G equals the number
of 1-vertex of H and the number of 1-vertex of G equals the
number of 0-vertex of H.
S
Proof. Since the vertices of G H S
are all the vertices of G
and H, the number of vertices of G H is 2n, where n is the
number of vertices of G .
S
For any friendly labeling f of G H, we have
vf (0) = vf (1) = n. Denote the number of 0-vertices in G and
H by vfG (0) and vfH (0), respectively. Also, denote the number
of 1-vertices in G and H by vfG (1) and vfH (1), respectively.
Sin-Min Lee and Hsin-hao Su
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Proof continued
On the
Balance Index
Sets of
Permutation
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Sin-Min Lee
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Thus, we have
vfG (0) + vfG (1) = n
and
vfH (0) + vfH (1) = n.
Also,
S by counting the number of 0-vertices and 1-vertices in
G H, we have
vfG (0) + vfH (0) = vf (0) = n
and
vfG (1) + vfH (1) = vf (1) = n.
These four equalities imply that vfG (1) = vfH (0) and
vfG (0) = vfH (1).
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Disjoint Union
On the
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Graphs
Sin-Min Lee
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Lemma
Let G and
S H be two graphs with the same number of vertices
and G H be the disjoint union of these two graphs. Let π be
any permutation of the set [n] where n is the number of
vertices of G . Then the balance index
S set BI(Perm(G , π, H)) is
equal to the balance index set BI(G H).
Proof. For any friendlySlabeling f of Perm(G , π, H), it is also
a friendly labeling of G H. Since f is friendly, we have
vf (0) = vf (1) = n in both graphs. Denote the number of
0-vertices in G and H by vfG (0) and vfH (0), respectively. Also,
denote the number of 1-vertices in G and H by vfG (1) and
vfH (1), respectively.
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Proof continued
On the
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Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
S
The only difference between Perm(G , π, H) and G H are the
permutation edges. We name the number of the permutation
edges which have one 0-vertex in G and one 1-vertex in H by
e01 . Similarly, we can define e00 , e10 and e11 in the same
manner. Obviously, we have
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e00 + e01 = vfG (0)
e10 + e11 = vfG (1)
e00 + e10 = vfH (0)
e01 + e11 = vfH (1).
The previous lemma implies vfG (1) = vfH (0). Thus, e00 = e11 .
Therefore, the permutation edges between two graphs do not
change the value of the balance index set.
2
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BI(Perm(G , π))
On the
Balance Index
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Graphs
Sin-Min Lee
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To calculate the balance index set BI(Perm(G , π)), by the
previous lemma, we only need to calculate the S
balance index
set of the disjoint union of two copiesSof G , G G . From now
on, we name theSfirst copy of G in G G , G T , and the second
copy of G in G G , G B .
Note that, by the first lemma we just proved, we have
vfT (1) = vfB (0) and vfT (0) = vfB (1) for any friendly labeling f .
From now on, we will omit the notation about the friendly
labeling f as long as the context is clear.
Sin-Min Lee and Hsin-hao Su
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Regular Graph
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
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Let REG (k) be the class of k-regular connected graph.
Theorem
For any G in REG(k) of order n, and for any permutation π of
[n], the balance index set
BI(Perm(G , π)) = {0}.
Proof. Let e T (0) and e T (1) be the number of 0- andS
1-vertices, respectively, in the first copy of G in the G G and
e B (0) and e B (1) be the number of 0- and 1-vertices,
S
respectively, in the second copy of G in the G G . Also, let
e T and e B be the number
of non-labeled edges in the first and
S
second copies of G G , respectively.
Sin-Min Lee and Hsin-hao Su
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Proof continued
On the
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Sets of
Permutation
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Sin-Min Lee
and
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6th Shanghai
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Since G T is a k-regular graph, every 0-vertex has k edges.
Thus, each 0-vertex is counted k times by the edges in G T
which has at least one 0-vertex. Every edge labeled 0 in G T
contributes two 0-vertices and every unlabeled edge in G T
contributes one 0-vertex. Therefore, we have
2e T (0) + e T
= kv T (0).
Similarly, by the same manner, we have three more equations:
Sin-Min Lee and Hsin-hao Su
2e T (1) + e T
= kv T (1)
2e B (0) + e B
= kv B (0)
2e B (1) + e B
= kv B (1).
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Proof continued
On the
Balance Index
Sets of
Permutation
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Sin-Min Lee
and
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6th Shanghai
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Since we have proven that v T (1) = v B (0) and v T (0) = v B (1),
we can conclude that
e T (0) − e T (1) = e B (1) − e B (0).
This completes the proof of the balance index set
BI(Perm(G , π)) = {0}
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Index Sets
on Combinatorics
of PermutationatGraphs
Shanghai Jiao Ton
Example: Perm(C6 , π)
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
The BI(Perm(C6 , π)) = {0} where
π(1) = 2, π(2) = 1, π(3) = 4, π(4) = 3, π(5) = 6, π(6) = 5.
6th Shanghai
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0
v1 0i 0
v2
0
0i
@
v4
0
0i
@
@
u1 1i
v3
0
0i
1
u2
1
0iv6
@
@
@
@i
1
v5
0
0i
1i
1
u3
@
@
@i
1
u4
1
1i
1
u5
@
@i
1 u6
1
|e(0) − e(1)| = 0
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
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Conference
Index Sets
on Combinatorics
of PermutationatGraphs
Shanghai Jiao Ton
Disjoint Union of Regular Graphs
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
The proof of the previous theorem actually tells us that
Corollary
For any G in REG(k) of order n, and for any permutation π of
[n], the balance index set of the disjoint union of two REG(k)s
is
[
BI(REG(k) REG(k)) = {0}.
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
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Index Sets
on Combinatorics
of PermutationatGraphs
Shanghai Jiao Ton
Stars
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
Let St(n) be the star graph with n vertices.
Theorem
For any permutation π, the balance index set
BI(Perm(St(n), π)) = {0, n − 2},
where n ≥ 2.
Proof. By the lemma, we only need to find the balance index
set of the disjoint union of two copies of St(n). Name the first
copy, StT (n) with its center c T , and the second copy, StB (n)
with its center c B .
S
With a friendly labeling in St(n) St(n), we have
v (0) = n = v (1) because the total number of the vertices is
even.
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
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Conference
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on Combinatorics
of PermutationatGraphs
Shanghai Jiao Ton
Proof continued
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
With loss of generosity, we can assume that c T is labeled by 0.
If c B is also labeled by 0, then every 0-vertex in the pendant
contributes an edge labeled 0 since it must connect to one of
the two centers which are labeled by 0. Thus, we have
e(0) = n − 2. But, none of the other edges are labeled by 1
since each edge which has 1-vertex in one end must connect to
one of the two centers which are labeled by 0. Therefore, the
balance index is n − 2.
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
the Balance
Conference
Index Sets
on Combinatorics
of PermutationatGraphs
Shanghai Jiao Ton
Proof continued
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
Assume that c B is labeled by 1. Let v T (0) be the number of
0-vertices in the first copy StT (n). Since c T is labeled by 0,
there are v T (0) − 1 0-vertices in the pendants of StT (n).
Therefore, there are v T (0) − 1 edges labeled by 0 in StT (n).
No edge in StT (n) is labeled by 1 since its center is labeled by
0. Also, because the center of StB (n) is labeled by 1, by the
same argument, in StB (n), there are v B (1) − 1 edges labeled
by 1 and no edges labeled by 0. From a previous lemma, we
B (1). Therefore, the number of edges
know that v T (0) = vS
labeled by 0 in St(n) St(n) is the same as the number of
edges labeled by 1. Thus, the balance index is 0.
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
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Shanghai Jiao Ton
Proof continued
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
These two cases tell us that the balance index set of
Perm(St(n), π) is BI(Perm(St(n), π)) = {0, n − 2}, where
n ≥ 2.
2
S
Note that St(1) is P2 . So, St(1) St(1) is two pairs of P2 . For
any friendly labeling, it is either both pairs are labeled by the
same number or both pairs are labeled by different number.
Therefore, the balance index set BI(Perm(St(1), π)) = {0}.
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
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Shanghai Jiao Ton
Example: Perm(St(5), π)
On the
Balance Index
Sets of
Permutation
Graphs
The BI(Perm(St(5), π)) = {0, 3}.
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
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Combinatorics
at
Shanghai Jiao
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University
u2
0i
u4
0i
0
u
0 T 3,
,
i
0
l
0 l
0 0i
u1 u5
0i
TT
u4
v4
TT
1i
1i1 v u
3
3 1
l
,
Ti
li
,
0
1
,
,
l i
1
0 l
T
i
1
1
1
TT v1 u1 u5
v5
i
i
0
1
|e(0) − e(1)| = 0
Sin-Min Lee and Hsin-hao Su
v2
0i
v2 u2
1i
1i
1
0
1
0
v4
1i v 3 0
l
li
0
, T0
,
i
1
TT v1
v5
0i
|e(0) − e(1)| = 3
6th
OnShanghai
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Shanghai Jiao Ton
Disjoint Union of Stars
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
The proof of the previous theorem actually tells us that
Corollary
Let St(n) be the star graph with n vertices. The balance index
set of the disjoint union of two St(n)s is
[
BI(St(n) St(n)) = {0, n − 2},
where n ≥ 2.
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
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Shanghai Jiao Ton
Pn
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
Lemma
The balance index set of Pn when n ≥ 4 is
(
{0, 1}
if n is even, and
BI(Pn ) =
{0, 1, 2} if n is odd.
Proof. Let the vertices of Pn be {v1 , v2 , . . . , vn }. If we add an
additional edge to connect v1 and vn , then we have a cycle Cn .
By the corollary 2.1 in Lee, Wang and Wen [9], the balance
index set of Cn is
(
{0} if n is even, and
BI(Cn ) =
{1} if n is odd.
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
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on Combinatorics
of PermutationatGraphs
Shanghai Jiao Ton
Proof continued
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
This additional edge contributes one edge labeled by 0 or 1 or
nothing. Thus, the balance index Pn is the balance index of
Cn ± 1 or Cn + 0. Therefore, when n is even, the balance index
set of Pn is {0, 1}. And, when n is odd, the balance index set
of Pn is {0, 1, 2}.
2
Note that BI(P3 ) = {0, 1}.
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
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Shanghai Jiao Ton
Perm(Pn , π)
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
Theorem
For any permutation π, the balance index set
(
{0, 1}
if n = 3, and
BI(Perm(Pn , π)) =
{0, 1, 2} if n ≥ 4.
Proof. By the lemma, we only need to find the balance index
set of the disjoint union of two copies of Pn . Name the first
copy, PnT with its vertices {t1 , t2 , . . . , tn }, and the second copy,
PnB with its vertices {b1 , b2 , . . . , bn }.
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
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Index Sets
on Combinatorics
of PermutationatGraphs
Shanghai Jiao Ton
Proof continued
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
S
If we add an additional edge t1 b1 into Pn Pn , we have a P2n .
As we just proved, the balance index set of P2n is
BI(P2n ) = {0, 1}.
If t1 and b1 are labeled by the different numbers, this
additional edge does not affect the balance index set. Thus, in
this case, the balance index set is still {0, 1}.
If t1 and b1 are labeled by the same numbers, without loss of
generosity, we can assume that t1 and b1 are both labeled by 0.
When n ≥ 4, we can have three different friendly labelings by
the following methods:
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
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Method 1
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
We label ti and bi by 1 if i is even and 0 is i is odd, where
1 ≤ i ≤ n. In this case, only the middle edge t1 b1 can
contribute an edge labeled by 0. Other edges are not labeled.
So, the balance index is e(0) − e(1) = 1 − 0 = 1. When we
remove
S the additional edge t1 b1 , we get the balance index of
Pn Pn of this labeling is 0.
···
Sin-Min Lee and Hsin-hao Su
0
1i 0i 1i 0i 1i 0i 0i 1i 0i 1i 0i 1i · · ·
t6 t5 t4 t3 t2 t1 b1 b2 b3 b4 b5 b6
6th
OnShanghai
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Method 2
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
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University
We label b2 , b3 , and t2 by 1, and t3 by 0. Moreover, we label
bi by 0 if i is even and 1 is i is odd when 4 ≤ i ≤ n and ti by 1
if i is even and 0 is i is odd when 4 ≤ i ≤ n. This friendly
labeling has only the edges t1 b1 labeled by 0 and the edges
b2 b3 labeled by 1. Thus, the balance index is
e(0) − e(1) = 1 − 1 = 0. When we remove the additional S
edge
t1 b1 which is labeled by 0, we get the balance index of Pn Pn
of this labeling is |e(0) − e(1)| = |0 − 1| = 1.
···
Sin-Min Lee and Hsin-hao Su
0
1
1i 0i 1i 0i 1i 0i 0i 1i 1i 0i 1i 0i · · ·
t6 t5 t4 t3 t2 t1 b1 b2 b3 b4 b5 b6
6th
OnShanghai
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Index Sets
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Shanghai Jiao Ton
Method 3
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
We label b2 , b3 , and t2 by 1, and t3 by 0. Moreover, we label
bi by 0 if i is even and 1 is i is odd when 4 ≤ i ≤ n and ti by 1
if i is even and 0 is i is odd when 4 ≤ i ≤ n. This friendly
labeling has only the edges t1 b1 labeled by 0 and the edges
b2 b3 labeled by 1. Thus, the balance index is
e(0) − e(1) = 1 − 1 = 0. When we remove the additional S
edge
t1 b1 which is labeled by 0, we get the balance index of Pn Pn
of this labeling is |e(0) − e(1)| = |0 − 1| = 2.
···
Sin-Min Lee and Hsin-hao Su
1
0
1
0i 1i 0i 1i 1i 0i 0i 1i 1i 0i 1i 0i · · ·
t6 t5 t4 t3 t2 t1 b1 b2 b3 b4 b5 b6
6th
OnShanghai
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on Combinatorics
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Shanghai Jiao Ton
Proof continued
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
Note that the first two cases are also true when n = 3. But, to
get the balance index 2 after removing the additional edge, we
need two edges labeled by 1. Since n = 3, we have only three
1-vertices. The only way to get two edges labeled by 1 is to
have 3 consecutive vertices labeled by 1. But, this cannot
happen since t1 and b1 are both labeled by 0 in the middle.
This completes our proof.
2
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
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Shanghai Jiao Ton
Example: Perm(P6 , π)
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
The BI(Perm(P6 , π)) = {0, 1, 2} where
π(1) = 2, π(2) = 3, π(3) = 4, π(4) = 5, π(5) = 6, π(6) = 1.
6th Shanghai
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t1
1
1i
t3
1
1i
t4
1
1i
t5
t6
1
i
1i
(1
(((
@
@
@(((@
(((@
(
(
@
@
@
@
(
(
(((@
(
@
@
@
(@
(
@
@
@
@
@i
0i
0i
0i
0i
0i
0
@
b1
Sin-Min Lee and Hsin-hao Su
t2
1
1i
0
b2
0
0
0
b3
b4
|e(0) − e(1)| = 0
b5
0
b6
6th
OnShanghai
the Balance
Conference
Index Sets
on Combinatorics
of PermutationatGraphs
Shanghai Jiao Ton
Example: Perm(P6 , π)
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
t1
1i 1
@
@1
t2
1i 1
b1
b2
t1
0i
t2
1i 1
@ 1
@
t3
1i 1
t4
1i 1
t5
t6
1i ((
0i
(
(((
@ 0 (@
@
((
((
@
@
@
(((@
(
(
(
(
@
@
@
@
(@
(
@
@
@
@
@i
0i
1i
0i
0i
0i
0
@
b3 0 b4 0
|e(0) − e(1)| = 1
t3
1i 1
t4
1i 1
b5
0
b6
t5
t6
i
1i ((
(0
(((
@ 0 (@
@
(
(( (@
(@
@
@
(
(
(
(( @
(
@
@
@
(@
(
@
@
@
@
@i
0i
1i
1i
0i
0i
0
@
b1
Sin-Min Lee and Hsin-hao Su
b2
1
b3
b4 0
|e(0) − e(1)| = 2
b5
0
b6
6th
OnShanghai
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Shanghai Jiao Ton
Disjoint Union of Pn
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
The proof of the previous theorem actually tells us that
Corollary
Let Pn be the path graph with n vertices. The balance index
set of the disjoint union of two Pn s is
(
[
{0, 1}
if n = 3, and
BI(Pn
Pn ) =
{0, 1, 2} if n ≥ 4.
Sin-Min Lee and Hsin-hao Su
6th
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Shanghai Jiao Ton
Perm(Cn (s), π)
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
Theorem
For any permutation π of [n], the balance index set
BI(Perm(Cn (s), π)) = {0, 1, 2},
for any n ≥ 4 and 3 ≤ t ≤ n − 2.
Proof. Again, by the lemma, we only need to find the balance
index set of the disjoint union of two copies of Cn (s). Name
the first copy, CnT (s) with its vertices {t1 , t2 , . . . , tn }, and the
second copy, CnB (s) with its vertices {b1 , b2 , . . . , bn }.
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
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Conference
Index Sets
on Combinatorics
of PermutationatGraphs
Shanghai Jiao Ton
Proof continued
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
If we remove the chord (t1 , ts ) and (b1 , bs ) from CnT (s) and
CnB (s), respectively, then the graph becomes the disjoint union
of two cycles CnT and CnB . By the Lemma
S 2.0 in Lee, Wang
and Wen [9], the balance index set of Cn is
(
P
[
{0} if
n is even, and
BI( Cn ) =
P
{1} if
n is odd.
Since bothScycles CnT and CnB are of order n, the balance index
set of CnT CnB is {0}.
Sin-Min Lee and Hsin-hao Su
6th
OnShanghai
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on Combinatorics
of PermutationatGraphs
Shanghai Jiao Ton
Proof continued
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
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University
For n ≥ 4, there are four cases to label the pairs (t1 , ts ) and
(b1 , bs ):
(1) both are labeled by the same number.
(2) one is labeled by 0 and another is labeled by 1.
(3) one is labeled by 0 or 1 and another is not labeled.
(4) both are not labeled.
Note that we can easily construct a friendly labeling of CnT and
CnB by filling out the rest of the vertices. So, these two chords
contribute nothing to the balance index set in the case II and
VI. With them, we have the balance index 1 in the case III and
2 in the case I. This completes our proof.
2
Sin-Min Lee and Hsin-hao Su
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Disjoint Union of Cn (s)
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
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at
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University
The proof of the previous theorem actually tells us that
Corollary
Let Cn (s) be the cycle with a chord. The balance index set of
the disjoint union of two Cn (s)s is
[
BI(Cn (s) Cn (s)) = {0, 1, 2},
for any n ≥ 4 and 3 ≤ t ≤ n − 2.
Sin-Min Lee and Hsin-hao Su
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Conjecture
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
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on
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at
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University
In this paper, we investigated the balance index sets of graphs
which are permutation graphs. We see that all of these sets
have values in arithmetic progressions. We proposed the
following conjecture.
Conjecture 1: The balance index set of any permutation
graph consists of arithmetic sequence.
Sin-Min Lee and Hsin-hao Su
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References
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
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at
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University
C.C. Chou and S.M. Lee, On the balance index sets of the
amalgamation of complete graphs and stars, manuscript.
F. Harary, Graph Theory, Reading, MA, Addison-Wesley, 1994.
Y.S. Ho, S.M. Lee, H.K. Ng and Y.H. Wen, On balancedness
of some families of trees, to appear in JCMCC. The Journal of
Combinatorial Mathematics and Combinatorial Computing, 38
(2001), 197-207.
R.Y. Kim, S.M. Lee and H.K. Ng, On balancedness of some
families of graphs, to appear in JCMCC.
H. Kwong and S.M. Lee, On balance index sets of chain sum
and amalgamation of generalized theta graphs, Congressus
Numerantium 187 (2007), 21-32.
Sin-Min Lee and Hsin-hao Su
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References
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
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University
H. Kwong, S.M. Lee and D.G. Sarvate, On balance index sets
of one-point unions of graphs, to appear in JCMCC.
A.N.T. Lee, S.M. Lee and H.K. Ng, On balance index sets of
graphs, to appear in JCMCC.
S.M. Lee, A. Liu and S.K. Tan, On balanced graphs,
Congressus Numerantium 87 (1992), 59-64.
S.M. Lee, Y.C. Wang and Y.H. Wen, On the balance index sets
of the (p, p + 1)-graphs, J. Combin. Math. Combin. Comput.,
62(2007), 193-216.
Sin-Min Lee and Hsin-hao Su
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References
On the
Balance Index
Sets of
Permutation
Graphs
Sin-Min Lee
and
Hsin-hao Su
6th Shanghai
Conference
on
Combinatorics
at
Shanghai Jiao
Tong
University
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