On Balanced Index Sets of
Disjoint Union Graphs
Sin-Min Lee
Department of Computer Science
San Jose State University
San Jose, CA 95192, USA
Hsin-Hao Su
Yung-Chin Wang *
Department of Mathematics
Stonehill College
Easton, MA 02357, USA
Department of Physical Therapy
Tzu-Hui Institute of Technology
Taiwan, Republic of China
40th SICCGC
March 2-6, 2009
Definition (A. Liu, S.K. Tan and S.M. Lee 1992)
Let G be a graph with vertex set V(G) and edge set
E(G).
A vertex labeling of G is a mapping f from V(G) into
the set {0, 1}.
For each vertex labeling f of G, define a partial edge
labeling f* of G from E(G) into the set {0, 1} as
following.
For each edge (u, v)E(G), where u, v V(G),
┌ 0, if f(u) = f(v) = 0,
f*(u,v) = ┤ 1, if f(u) = f(v) = 1,
└ undefined, if f(u) ≠ f(v) .
Definition (A. Liu, S.K. Tan and S.M. Lee 1992)
A graph G is said to be a balanced graph
or G is balanced if there is a vertex
labeling f of G satisfying |vf(0) – vf(1)| ≤ 1
and |ef*(0) – ef*(1) | ≤ 1.
Definition (A.N.T. Lee, S.M. Lee, H.K. Ng 2008)
The balance index set of a graph G, BI(G),
is defined as {|ef*(0) – ef*(1)| : the vertex
labeling f is friendly}.
Example. BI(K3,3) = {0}
Example. BI(DS(2,2)) = {0,2},
BI(DS(3,3)) = {0,3}.
Theorem (Kwong, Lee, Lo, Wang 2008)
Let G be a k-regular graph G of order p.
Then
┌{0} if p is even,
BI(G) =┤
└{k/2} if p is odd.
Permutation Graphs
Let  be a permutation of the set [n]= {1,2,…,n}.
For a graph G of order n, the -permutation
graph of G is the disjoint union of two copies of
G, namely, GT and GB, together with the edges
joining the vertex vi of GT with v(i) of GB.
Theorem (Lee & Su)
Let G and H be two graphs with the same
number of vertices and G∪H be the
disjoint union of these two graphs. Let 
be any permutation between the vertex
sets of G and H. Then, the balance index
set
BI(Perm(G,,H)) = BI(G∪H).
Theorem (Lee & Su)
Let G and H be two graphs with the same
order, if both of them are k-regular graphs,
then
BI(G∪H)={0}.
Example
Let G and H be two 4-regular graphs as below,
then BI(G ∪ H)={0}.
Lemma
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-vertices of
G equals the number of 1-vertices of H
and the number of 1-vertices of G equals
the number of 0-vertices of H, i.e.,
vG(1) = vH(0) and vG(0) = vH(1).
Theorem
For any G in REG(s) and H in REG(t) of
order n and any friendly labeling f on G∪H,
we have
2( e(0) - e(1) ) = ( s - t )( vG(0) - vH(0) )
= ( s - t )( 2vG(0) - n )
= ( s - t )( n - 2vH(0) )
Theorem
Let G and H be two graphs with the same
order n, if G is a k-regular graph and H is
an h-regular graph, k≠h, then
1. { 0, |s-t|, 2|s-t|, 3|s-t|, …, (n/2)|s-t| }, if n is
even,
2. { |(s-t)/2|, 3|(s-t)/2|, 5|(s-t)/2|, …, n|(s-t)/2| },
if n is odd.
Example
BI(C4 ∪ K4)={0,1,2}
Theorem
BI(Cn∪Pn))={0,1}.
Example. BI(C6∪P6)={0,1}
Theorem
BI(Cn∪St(n-1))={0,1,2,…,n-2}.
Example. BI(C4∪St(3))={0,1,2}
Theorem
BI(Pn∪St(n-1))={0,1,2,…,n-2}
Example. BI(P6∪St(5))={0,1,2,3,4}
Theorem.
Let BI(SP(2[n])) be the spider.
We have
1. BI(SP(2[n])) = {0,1,…,n}
2. BI(SP(2[n]) ∪ SP (2[n]) )={0,1,2,…,2(n-1)}
SP(2[3])
Theorem.
Let CT(1[n]) be the corona of a path Pn.
We have
1. BI(CT(1[n]) )={0,1,2,…,n-1}
2. BI(CT(1[n]) ∪ CT(1[n]) )={0,1,2,…,2(n-1)}
CT(1[5])
Theorem
Let DS(m, n) be the double star. We have
1. {(n – m)/2, (n + m)/2}, if m + n is even,
2. {(n – m – 1)/2, (n – m + 1)/2, (n + m –
1)/2, (n + m + 1)/2}, if m + n is odd.
Unsolved Problem
For what m,n, BI(DS(m,n) )  DS(m,n)))
forms arithmetic progression?