On Balanced Index Sets of Disjoint Union Graphs

Sin-Min LeeDepartment of Computer Science

San Jose State UniversitySan Jose, CA 95192, USA

Hsin-Hao SuDepartment of Mathematics

Stonehill CollegeEaston, MA 02357, USA

Yung-Chin Wang *Department of Physical TherapyTzu-Hui Institute of Technology

Taiwan, Republic of China

40th SICCGCMarch 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 disjoi∪nt 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 K∪ 4)={0,1,2}

Theorem

BI(Cn∪ Pn))={0,1}.

Example. BI(C6 P∪ 6)={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?