On Balanced Index Sets of Disjoint Union Graphs

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 Department of Mathematics Stonehill College Easton, MA 02357, USA Yung-Chin Wang * 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 balancedgraph 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. Ng2008) 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 Gis 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 • { 0, |s-t|, 2|s-t|, 3|s-t|, …, (n/2)|s-t| }, if n is even, • { |(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 • BI(SP(2[n])) = {0,1,…,n} • 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 • BI(CT(1[n]) )={0,1,2,…,n-1} • 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 • {(n – m)/2, (n + m)/2}, if m + n is even, • {(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?

On Balanced Index Sets of Disjoint Union Graphs