IJCTInternational Journal of Combinatorics1687-91711687-9163Hindawi Publishing Corporation28438310.1155/2012/284383284383Research ArticleTotal Vertex Irregularity Strength of the Disjoint Union of Sun GraphsSlamin1Dafik2WinnonaWyse2YusterR.1Information System Study ProgramUniversity of Jember, Jember 68121Indonesiaunej.ac.id2Mathematics Education Study ProgramUniversity of Jember, Jember 68121Indonesiaunej.ac.id2012382011201211012011310320112012Copyright © 2012 Slamin et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A vertex irregular total k-labeling of a graph G with vertex set V and edge set E is an assignment of positive integer labels {1,2,,k} to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of G, denoted by tvs(G) is the minimum value of the largest label k over all such irregular assignment. In this paper, we consider the total vertex irregularity strengths of disjoint union of s isomorphic sun graphs, tvs(sMn), disjoint union of s consecutive nonisomorphic sun graphs, tvs(i=1sMi+2), and disjoint union of any two nonisomorphic sun graphs tvs(MkMn).

1. Introduction

Let G be a finite, simple, and undirected graph with vertex set V and edge set E. A vertex irregular total k-labeling on a graph G is an assignment of integer labels {1,2,,k} to both vertices and edges such that the weights calculated at vertices are distinct. The weight of a vertex vV in G is defined as the sum of the label of v and the labels of all the edges incident with v, that is,wt(v)=λ(v)+uvEλ(uv).

The notion of the vertex irregular total k-labeling was introduced by Bača et al. . The total vertex irregularity strength of G, denoted by tvs(G), is the minimum value of the largest label k over all such irregular assignments.

The total vertex irregular strengths for various classes of graphs have been determined. For instances, Bača et al.  proved that if a tree T with n pendant vertices and no vertices of degree 2, then (n+1)/2tvs(T)n. Additionally, they gave a lower bound and an upper bound on total vertex irregular strength for any graph G with v vertices and e edges, minimum degree δ and maximum degree Δ, (|V|+δ)/(Δ+1)tvs(G)|V|+Δ-2δ+1. In the same paper, they gave the total vertex irregular strengths of cycles, stars, and complete graphs, that is, tvs(Cn)=(n+2)/3, tvs(K1,n)=(n+1)/2 and tvs(Kn)=2.

Furthermore, the total vertex irregularity strength of complete bipartite graphs Km,n for some m and n had been found by Wijaya et al. , namely, tvs(K2,n)=(n+2)/3 for n>3, tvs(Kn,n)=3 for n3, tvs(Kn,n+1)=3 for n3, tvs(Kn,n+2)=3 for n4, and tvs(Kn,an)=n(a+1)/(n+1) for all n and a>1. Besides, they gave the lower bound on tvs(Km,n) for m<n, that is, tvs(Km,n)max{(m+n)/(m+1),(2m+n-1)/n}. Wijaya and Slamin  found the values of total vertex irregularity strength of wheels Wn, fans Fn, suns Mn and friendship graphs fn by showing that tvs(Wn)=(n+3)/4, tvs(Fn)=(n+2)/4, tvs(Mn)=(n+1)/2, tvs(fn)=(2n+2)/3.

Ahmad et al.  had determined total vertex irregularity strength of Halin graph. Whereas the total vertex irregularity strength of trees, several types of trees and disjoint union of t copies of path had been determined by Nurdin et al. . Ahmad and Bača  investigated the total vertex irregularity strength of Jahangir graphs Jn,2 and proved that tvs(Jn,2)=(n+1)/2, for n4 and conjectured that for n3 and m3, tvs(Jm,n)max{n(m-1)+23,nm+24}. They also proved that for the circulant graph, tvs(Cn(1,2))=(n+4)/5, and conjectured that for the circulant graph Cn(a1,a2,,am) with degree at least 5, 1ain/2, tvs(Cn(a1,a2,,am))=(n+r)/(1+r).

A sun graph Mn is defined as the graph obtained from a cycle Cn by adding a pendant edge to every vertex in the cycle. In this paper, we determine the total vertex irregularity strength of disjoint union of the isomorphic sun graphs tvs(sMn), disjoint union of consecutive nonisomorphic sun graphs tvs(i=1sMi+2) and disjoint union of two nonisomorphic sun graphs tvs(MkMn), as described in the following section.

2. Main Results

We start this section with a lemma on the lower bound of total vertex irregularity strength of disjoint union of any sun graphs as follows.

Lemma 2.1.

The total vertex irregularity strength of disjoint union of any sun graphs is tvs(i=1sMni)((i=1sni)+1)/2,  for  s1, ni+1ni, and 1is.

Proof.

The disjoint union of the isomorphic sun graphs i=1sMni has i=1sni vertices ui,j of degree 1 and i=1sni vertices vi,j of degree 3. Note that the smallest weight of vertices of i=1sMni must be 2. It follows that the largest weight of i=1sni vertices of degree 1 is at least (i=1sni)+1 and of i=1sni vertices of degree 3 is at least 2(i=1sni)+1. As a consequence, at least one vertex ui,j or one edge incident with ui,j has label at least ((i=1sni)+1)/2. Moreover, at least one vertex vi,j or one edge incident with vi,j has label at least (2(i=1sni)+1)/4. Then tvs(i=1sMni)max{(i=1sni)+12,2(i=1sni)+14}. Because of (i=1sni)+12=2(i=1sni)+14, then tvs(i=1sMni)(i=1sni)+12.

We now present a theorem on the total vertex irregularity strength of disjoint union of the isomorphic sun graphs tvs(sMn) as follows.

Theorem 2.2.

The total vertex irregularity strength of the disjoint union of isomorphic sun graphs is tvs(sMn)=(sn+1)/2, for s1 and n3.

Proof.

Using Lemma 2.1, we have tvs(sMn)(sn+1)/2. To show that tvs(sMn)(sn+1)/2, we label the vertices and edges of sMn as a total vertex irregular labeling. Suppose the disjoint union of the isomorphic sun graphs sMn has the set of vertices V(sMn)={ui,j1is,  1jn}{vi,j1is,  1jn}   and the set of edges E(sMn)={ui,jvi,j1is,  1jn}{vi,jvi,j+11is,  1jn}. The labels of the edges and the vertices of sMn are described in the following formulas: λ(ui,j)={1for  i=1,2,,s-12;  j=1,2,,nand  i=s+12,  j=1,2,,(sn+12-ns-12)1+j+(i-1)n-sn+12for  other  i,j,λ(vi,j)={1for  i=1,2,,s-12;  j=1,2,,nand  i=s+12,  j=1,2,,(sn+12-ns-12)1+j+(i-1)n-sn+12for  other  i,j,λ(ui,jvi,j)={j+(i-1)nfor  i=1,2,,s-12;  j=1,2,,nand  i=s+12,  j=1,2,,(sn+12-ns-12)sn+12for  other  i,j,λ(vi,jvi,j+1)=sn+12,for  i=1,2,,s,  j=1,2,,n. The weights of the vertices ui,j and vi,j of sMn are wt(ui,j)=1+j+(i-1)n,for  i=1,2,,s,  j=1,2,,n,wt(vi,j)=1+j+(i-1)n+2sn+12,for  i=1,2,,s,  j=1,2,,n. It is easy to see that the weights calculated at vertices are distinct. So, the labeling is vertex irregular total. Therefore tvs(sMn)=(sn+1)/2 for s1 and n3.

Figure 1 illustrates the total vertex irregular labeling of the disjoint union 5 copies sun graphs M5.

Vertex irregular total 13 labelings of 5M5.

If we substitute s=1 into the theorem above, we obtain a result that has been proved by Wijaya and Slamin  as follows.

Corollary 2.3.

The total vertex irregularity strength of sun graph tvs(Mn)=(n+1)/2, for s=1 and n3.

The following theorem shows the total vertex irregularity strength of disjoint union of nonisomorphic sun graphs with consecutive number of pendants.

Theorem 2.4.

The total vertex irregularity strength of disjoint union of consecutive nonisomorphic sun graphs is tvs(i=1sMi+2)=(s(s+5))/4, for s1.

Proof.

Using Lemma 2.1, we have tvs(i=1sMi+2)(s(s+5)+2)/4. To show that tvs(i=1sMi+2)(s(s+5)+2)/4, we label the vertices and edges of i=1sMi+2 as a total vertex irregular labeling. Suppose the disjoint union of the nonisomorphic sun graphs with consecutive number of pendants i=1sMi+2 has the set of vertices V(i=1sMi+2)={u1,1,u1,2,u1,3,u2,1,u2,2,u2,3,u2,4,,us,1,us,2,,us,s+2,v1,1,v1,2,v1,3,v2,1,v2,2,v2,3,v2,4,,vs,1,vs,2,vs,s+2} and the set of edges E(i=1sMi+2)={u1,1v1,1,,u1,3v1,3,u2,1v2,1,,u2,4v2,4,,us,1vs,1,us,2vs,2,,us,s+2vs,s+2}{v1,1v1,2,,v1,3v1,1,v2,1v2,2,,v2,4v2,1,,vs,1vs,2,,vs,s+2vs,1}. The labels of the edges and the vertices of i=1sMi+2 are described in the following formulas: λ(ui,j)={1for  i=1,2,,(2s3-1),  j=1,2,,i+2,i=2s3;j=1,2,,(s(s+5)+24-(2s/3-1)(2s/3+4)2)1+j+(i2+3i-42)-s(s+5)+24for  other  i,j,λ(vi,j)={1for  i=1,2,,(2s3-1),  j=1,2,,i+2,i=2s3;j=1,2,,(s(s+5)+24-(2s/3-1)(2s/3+4)2)1+j+(i2+3i-42)-s(s+5)+24for  other  i,j,λ(ui,jvi,j)={j+(i2+3i-42)for  i=1,2,,(2s3-1),  j=1,2,,i+2,i=2s3;j=1,2,,(s(s+5)+24-(2s/3-1)(2s/3+4)2)s(s+5)+24for  other  i,j,λ(vi,jvi,j+1)=s(s+5)+24,for  i=1,2,,s,  j=1,2,,i+2. The weights of the vertices ui,j and vi,j of i=1sMi+2 are wt(ui,j)=i2+3i+2j-22,for  i=1,2,,s,  j=1,2,,i+2,wt(vi,j)=i2+3i+2j-22+2s(s+5)+24for  i=1,2,,s,  j=1,2,,i+2. It is easy to see that the weights calculated at vertices are distinct. So, the labeling is vertex irregular total. Therefore tvs(i=1sMi+2)=(s(s+5)+2)/4 for s1 dan n3.

Figure 2 illustrates the vertex irregular total 10 labelings of the disjoint union 4 consecutive nonisomorphic sun graphs M3M4M5M6.

Vertex irregular total 10 labelings of M3M4M5M6.

Finally, we conclude this section with a result on the total vertex irregularity strength of disjoint union of two nonisomorphic sun graphs as follows.

Theorem 2.5.

The total vertex irregularity strength of disjoint union of two nonisomorphic sun graphs is tvs(MkMn)=(k+n+1)/2, for n>k3.

Proof.

Using Lemma 2.1, we have tvs(i=1sMi+2)(k+n+1)/2. To show that tvs(MkMn)(k+n+1)/2, we label the vertices and edges of MkMn as a vertex irregular total k-labeling. Suppose the disjoint union of the nonisomorphic sun graphs with different pendant MkMn has the set of vertices V(sMn)={u1,1,u1,2,,u1,n,u2,1,u2,2,,u2,n,v1,1,v1,2,,v1,n,v2,1,v2,2,,v2,n} and the set of edges E(sMn)={u1,1v1,1,u1,2v1,2,,u1,nv1,n,u2,1v2,1,u2,2v2,2,,u2,nv2,n}{v1,1v1,2,v1,2v1,3,,v1,nv1,1,v2,1v2,2,v2,2v2,3,,v2,nv2,1}. The labels of the edges and the vertices of MkMn are described in the following formulas: λ(ui,j)={1for  i=1;  j=1,2,,k,i=2;  j=1,2,,(k+n+12-1)1+j+(i-1)k+(k+n+12)for  other  i,j,λ(vi,j)={1for  i=1;  j=1,2,,k,i=2;  j=1,2,,(k+n+12-1)1+j+(i-1)k+(k+n+12)for  other  i,j,λ(ui,jvi,j)={j+(i-1)kfor  i=1;  j=1,2,,k,i=2;  j=1,2,,(k+n+12-1)k+n+12for  other  i,j,λ(vi,jvi,j+1)=k+n+12,  for  i=1;  j=1,2,,k,  i=2;  j=1,2,,n. The weights of the vertices ui,j and vi,j of MkMn are wt(ui,j)=1+j+(i-1)k,for  i=1,2,  j=1,2,,n,wt(vi,j)=1+j+(i-1)k+2(k+n+12),for  i=1,2,  j=1,2,,n. It is easy to see that the weights calculated at vertices are distinct. So, the labeling is vertex irregular total. Therefore tvs(MkMn)=(k+n+1)/2 for n>k3.

Figure 3 illustrates the vertex irregular total 6 labelings of the disjoint union of 2 nonisomorphic sun graphs M4M6.

Vertex irregular total 6 labelings of M4M6.

3. Conclusion

We conclude this paper with the following conjecture for the direction of further research in this area.

Conjecture 1.

The total vertex irregularity strength of disjoint union of any sun graphs is tvs(i=1sMni)=((i=1sni)+1)/2, for s1, ni+1ni and 1is.

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