Total vertex irregularity strength of corona product of some graphs

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1 Journal of Algorithms and Computation journal homepage: Total vertex irregularity strength of corona product of some graphs P. Jeyanthi and A. Sudha 2 Research Centre, Department of Mathematics, Govindammal Aditanar College for Women, Tiruchendur-628 2, Tamil Nadu, India. 2 Department of Mathematics, Wavoo Wajeeha Women s College of Arts and Science, Kayalpatnam ,Tamil Nadu, India. ABSTRACT 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,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 study the total vertex irregularity strength for n, m 2,P n K,P n K 2,C n K 2,L n K,CL n K,P 2 C n,p n K m,c n K m ARTICLE INFO Article history: Received 22, August 206 Received in revised form 8, November 206 Accepted December 206 Available online 2, December 206 Keyword: irregularity strength; total vertex irregularity strength; vertex irregular total labeling; corona product of path and cycle; path and complete graph; ladder and complete graph. graph. AMS subject Classification: 0C78. Corresponding author: P. Jeyanthi. jeyajeyanthi@rediffmail.com sudhathanalakshmi@gmail.com Journal of Algorithms and Computation 48 (206) PP

2 28 Jeyanthi / Journal of Algorithms and Computation 48 (206) PP Introduction As a standard notation, assume that G = (V,E) is a finite,simple and undirected graph with p vertices and q edges. A labeling of a graph is any mapping that sends some set of graph elements to a set of numbers(usually positive integers). If the domain is the vertex -set (or) the edge- set,the labeling are called respectively vertex labeling (or) edge labeling. If the domain is V E then we call the labeling a total labeling. In many cases it is interesting to consider the sum of all labels associated with a graph element. This will be called the weight of element. The graph labeling has caught the attention of many authors and many new labeling results appear every year. This popularity is not only due to the mathematical challenges of graph labeling, but also for the wide range of its application for instance X-ray, crystallography, coding theory, radar, astronomy, circuit, design, network design and communication design. Chartrand et al. [7] introduced labelings of the edges of a graph G with positive integers such that the sum of the labels of edges incident with a vertex is different for all the vertices. Such labelings were called irregular assignments and the irregularity strength s(g) of a graph G is known as the minimum k for which G has an irregular assignment using labels at most k. The irregularity strength s(g) can be interpreted as the smallest integer k for which G can be turned into a multigraph G by replacing each edge by a set of at most k parallel edges, such that the degrees of the vertices in G are all different.finding the irregularity strength of a graph seems to be hard even for graphs with simple structure in [6, 6]. Karonski et al. [9] conjectured that the edges of every connected graph of order at least can be assigned labels from,2,}such that for all pairs of adjacent vertices the sums of the labels of the incident edges are different. Motivated by irregular assignments Bača et al. [] defined a labeling f : V(G) E(G),2,...,k}tobeavertexirregulartotalk-labelingifforeverytwodifferentverticesxandy thevertex-weightswt f (x) wt f (y)wherethevertex-weightwt f (x) = f(x)+ xy E f(xy). A minimum k for which G has a vertex irregular total k-labeling is defined as the total vertex irregularity strength of G and denoted by tvs(g). It is easy to see that irregularity strengths(g)ofagraphgisdefinedonlyforgraphscontainingatmostoneisolatedvertex and no connected component of order 2. On the other hand, the total vertex irregularity strength tvs(g) is defined for every graph G. If an edge labeling f : E,2,...,δ(G)} provides the irregularity strength s(g), then we extend this labeling total labeling φ in such a way φ(xy) = f(xy) for everyxy E(G), φ(x) = for every x V(G). Thus,the total labeling φ is a vertex irregular total labeling and for graphs with no component of order 2 has tvs(g) s(g). Nierhoff [0] proved that for all (p,q)-graphs G with no component of order at most 2 and G K the irregularity strength s(g) p. From this result it follows that

3 29 Jeyanthi / Journal of Algorithms and Computation 48 (206) PP tvs(g) p. () In [] several bounds and exact values of tvs(g) were determined for different types of graphs(in particular for stars, cliques and prisms). Among others, the authors proved that for every (p,q)- graph G with minimum degree δ = δ(g)and maximum degree = (G), p+δ(g) tvs(g) p+ (G) 2δ(G)+. (2) (G)+ In the case of r-regular graphs (2) gives p+r tvs(g) p r+. () r + For graphs with no component of order 2, Bača et al. [] strengthened also these upper bounds, proving that tvs(g) p. These results were then improved by p 2 (G)+ Przybylo in [4] for sparse graphs and for graphs with large minimum degree. Ahmad et al [,] determined an exact value of the total vertex irregularity strength for wheel related graphs and cubic graphs. Wijaya et al. [8] determined an exact value of the total vertex irregularity strength for complete bipartite graphs. Wijaya et al. [7] found the exact values of tvs for wheels, fans, suns and friendship graphs. Nurdin et al. [] proved the following lower bound of tvs for any graph G. Theorem: [] Let G be a connected graph having n i vertices of degree i(i = δ,δ +,δ+2,..., ) whereδ and are theminimum and themaximum degreeof G, respectively. Then δ +nδ δ +nδ +n δ + (n i ) δ+ i=δ tvs(g) max,,...,. (4) δ + δ +2 + Also [] Nurdin et al. posed the following conjecture. Conjecture:.2 [] Let G be a connected graph having n i vertices of a degree i(i = δ,δ +,δ + 2,..., ) whereδ and are the minimum and the maximum degree of G, respectively.then δ +nδ δ +nδ +n δ + (n i ) δ+ i=δ tvs(g) = max,,...,. () δ + δ +2 + Conjecture.2 has been verified by several authors for several families of graphs. For a regular Hamiltonian (p,q) graph G, it was showed in [] that tvs(g) p+2. Thus for cycle C p we have that tvs(g) = p+2. In [,2, ], Nurdin et al. found the exact values of total vertex irregularity strength of trees, several types of trees and disjoint union of

4 0 Jeyanthi / Journal of Algorithms and Computation 48 (206) PP t copies of path. Slamin et al. [] determined the total vertex irregularity strength of disjoint union of sun graphs. In [2] Ahmad, Bača and Numan determined the total vertex irregularity strength of disjoint union of friendship graphs. In [4] Ashfaq Ahmad, Syed Ahtsham ul Haq Bokhary, Roslan Hasni and Slamin found the exact value of the total vertex irregularity strength of ladder related graphs. Definition.. A corona product of two graphs G and H, denoted by G H, is the graph that is obtained by placing a copy of G and V(G) copies of H so that all vertices in the same copy of H are joined with exactly one vertex of G, while each vertex of G is joined to exactly one copy of H. Definition.2. The ladder L n is the planar grid P n X P 2. 2 Main Results In this paper we determine exact values of the total vertex irregularity strength for n,m 2,P n K,P n K 2,C n K 2,L n K,CL n K,P 2 C n,p n K m,c n K m. Theorem 2.. tvs(p n K ) = n+ 2,n. Proof. Let V(P n K ) = u i,v i : i n} and E(P n K ) = u i v i,v i v i+ : i n}. Letk = n+,thenfrom(4)itfollowsthat,tvs(pn K 2 ) max n+ 2, n+, 2n+ } 4 = n+ 2. That is tvs(pn K ) k. To prove the reverse inequality,we define a function f from V E to,2,,...,k} in the following way., if i k f(u i ) = +i k, if k + i n; We observe that for i n, k, if i = 2k n, if 2 i k f(v i ) = k n+i, if k + i n 2 n k, if i = n n+ 2k, if i = n; i, if i k f(u i v i ) = k, if k + i n; n k +, if i n 2 f(v i v i+ ) = k, if i = n. wt(u i ) = i+,

5 Jeyanthi / Journal of Algorithms and Computation 48 (206) PP n+2, if i = n+2+i, if 2 i k wt(v i ) = n+2+i, if k + i n 2 2n+, if i = n n+, if i = n. It is easy to check that the weights of the vertices are distinct.this labeling construction shows that tvs(p n K ) k. Combining this with the lower bound, we conclude that tvs(p n K ) = k. Figure shows the vertex irregular total labeling of P 8 K Figure : tvs(p 8 K ) = Theorem 2.2. tvs(p n K 2 ) =,n. Proof. Let V(P n K 2 ) = v i,a i,b i : i n } and E(P n K 2 ) = v i v i+,v i a i,v i b i,a i b i : i n}. Let k =, then from (4) it follows that, tvs(p n K 2 ) max, 2n+4 4, n+2 } =.Thatistvs(Pn K 2 ) = k. To prove the reverse inequality, we define a function f from V E,2,,...,k} in the following way. f(a i ) =, i n; f(v i b i ) = k, i n; f(v i a i ) =, if i k +i k, if k + i n; f(b i n+2 k, if i k ) = n+2 2k +i, if k + i n; f(a i b i i, if i k ) = k, if k + i n;

6 2 Jeyanthi / Journal of Algorithms and Computation 48 (206) PP k, if i = 2k +4 2k, if i = 2 f(v i ) = k +i, if i k 2k, if k + i n n+ k, if i = n; We observe that, f(v i v i+ ) =, if i = k, if 2 i n. wt(a i ) = 2+i, i n; wt(b i ) = n+2+i, i n; 2n+, if i = 2n+, if i = 2 wt(v i ) = 2n++i, if i k 2n++i, if k + i n 2n+4, if i = n. It is easy to check that the weights of the vertices are distinct. This labeling construction shows that tvs(p n K 2 ) k. Combining this with the lower bound, we conclude that tvs(p n K 2 ) = k. Figure 2 shows the vertex irregular total labeling of P 6 K Figure 2: tvs(p 6 K 2 ) = Theorem 2.. tvs(c n K 2 ) =, n. Proof. Let V(C n K 2 ) = v i,a i,b i : i n} and E(C n K 2 ) = v i v i+,v i a i,v i b i,a i b i : i n} with indices taken modulo n. Let k =, then from (4) it follows that, tvs(cn K 2 ) max, 2n+4 4, n+2 }

7 Jeyanthi / Journal of Algorithms and Computation 48 (206) PP =. That is tvs(cn K 2 ) = k. To prove the reverse inequality, we define a function f from V E to,2,,...,k} in the following way. f(a i, if i k ) = 2i 2k +, if k + i n; f(v i v i+ ) = k, i n. We observe that, wt(a i ) = 2i+, i n; wt(b i ) = 2i+2, i n; wt(v i ) = k +2+i, i n., if i k f(b i ) = 2, if i = k 2i 2k +2, if k + i n; f(v i a i i, if i k ) = k, if k + i n; f(v i b i +i, if i k ) = k, if k + i n; f(a i b i i, if i k ) = k, if k + i n; +k i, if i k f(v i ) = 2, if i = k 2+i k, if k + i n; It is easy to check that the weights of the vertices are distinct. This labeling construction shows that tvs(c n K 2 ) k. Combining this with the lower bound, we conclude that tvs(c n K 2 ) = k. Figure shows the vertex irregular total labeling of C 6 K 2.

8 4 Jeyanthi / Journal of Algorithms and Computation 48 (206) PP Figure : tvs(c 6 K 2 ) = Theorem 2.4. tvs(l n K ) = n+,n. Proof. Let V(L n K ) = v i,u i,a i,b i : i n} and E(L n K ) = v i v i+,u i u i+,v i u i,v i a i,v i b i : i n}, then from (4) it follows that, tvs(l n K ) max 2n+ 2, 2n+ 4, n+ } = 2n+ 2 = n +. That is tvs(ln K ) n +. To prove the reverse inequality, we define a function f from V E to,2,,...,n+} in the following way. n, if i n f(v i ) = 2, if i = n; +n, if i =,n f(u i ) =, if 2 i n ; f(v i v i+ ) =, i n; f(u i u i+ ) = n, i n; f(a i ) =, i n; f(b i ) = n+, i n; f(v i a i ) = f(u i b i ) = i, i n; f(v i u i ) = n, i n. We observe that, wt(a i ) = +i, i n; wt(b i ) = n++i, i n;

9 Jeyanthi / Journal of Algorithms and Computation 48 (206) PP n+2, if i = wt(u i ) = n++i, if 2 i n 4n+, if i = n;, if i = wt(v i ) = +i, if 2 i n 2n+, if i = n. It is easy to check that the weights of the vertices are distinct. This labeling construction shows that tvs(l n K ) n+. Combining this with the lower bound, we conclude that tvs(l n K ) = n+. Figure 4 shows the vertex irregular total labeling of L K Figure 4: tvs(l K ) = 6 Theorem 2.. tvs(cl n K ) = n+,n. Proof. Let V(CL n K ) = v i,u i,a i,b i : i n} and E(CL n K ) = v i v i+,u i u i+,v i u i,v i a i,v i b i : i n}, with indices taken modulo n. Then from (4) it follows that, tvs(cl n K ) max 2n+ 2, 4n+ } = 2n+ 2 = n+. That is tvs(cl n K ) n+. To prove the reverse inequality, we define a function f from V E to,2,,...,n+} in the following way. f(v i ) = n, i n; f(u i ) =, i n; f(v i v i+ ) =, i n; f(u i u i+ ) = n, i n; f(a i ) =, i n; f(b i ) = n+, i n; f(v i a i ) = f(u i b i ) = i, i n; f(v i u i ) = n, i n.

10 6 Jeyanthi / Journal of Algorithms and Computation 48 (206) PP We observe that, wt(a i ) = +i, i n; wt(b i ) = n++i, i n; wt(v i ) = 2n++i, i n; wt(u i ) = n++i, i n. It is easy to check that the weights of the vertices are distinct. This labeling construction shows that tvs(cl n K ) n+. Combining this with the lower bound, we conclude thattvs(cl n K ) = n+.figureshowsthevertexirregulartotallabelingofcl K Figure : tvs(cl K ) = 6 Theorem 2.6. tvs(p 2 C n ) = 2n+ 4,n. Proof. LetV(P 2 C n ) = c j,v j i : i n, j 2} and E(P 2 C n ) = v j i,vj i+,c jv j i,: i n, j 2}. Let k = 2n+, then from (4) it follows that,tvs(p 2 C n ) max 2n+ 4, n+4 } n+2 = 2n+ 4 = k. That is tvs(p2 C n ) 2n+ 4 = k. To prove the reverse inequality, we define a function f from V E to,2,,...,k} in the following way. For i n, j 2, f(v j i ) =, if i k +i k, if k + i n;

11 7 Jeyanthi / Journal of Algorithms and Computation 48 (206) PP f(v iv i+) =, i n; f(v 2 iv 2 i+) = k, i n; f(c ) = k, f(c 2 ) = k, f(c c 2 ) =. f(c j v j i ) = i, if i k k, if k + i n; We observe that, wt(vi) = +i, i n; wt(vi) 2 = 2k ++i, i n; wt(c ) = k(m+ k)+ k (i), i= wt(c 2 ) = k(m+ k)+ k (i+). i= It is easy to check that the weights of the vertices are distinct. This labeling construction shows that tvs(p 2 C n ) k. Combining this with the lower bound, we conclude that tvs(p 2 C n ) = k. Figure 6 shows the vertex irregular total labeling of P 2 C Figure 6: tvs(p 2 C 8 ) = Theorem 2.7. tvs(c n K m ) = +nm 2,n,m 2. Proof. Let V(C n K m ) = v i,a j i : i n, j m} and E(C n K m ) = v i v i+,v i a j i : i n, j m} with indices taken modulo n. Let k = +nm 2, then from (4) it follows that, tvs(c n K m ) max +nm 2, +nm+n } n+ = +nm 2 = k. That is tvs(c n K m ) k. To prove the reverse inequality, we define a function f from V E to,2,,...,k} in the following way. f(v i ) = i, i n; f(v i v i+ ) = k, i n; f(a i) = i, i n; f(v i a j i ) = min+(n )(j ),k}; i n, j m.

12 8 Jeyanthi / Journal of Algorithms and Computation 48 (206) PP For i n,2 j m,+(n )(j ) < k f(a j i ) = i+j. For i n,2 j m,+(n )(j ) k f(a j i ) = (j )n+ k +i. We observe that, wt(a j i ) = (j )n++i, i n, j m. Also the weights of v i s are distinct.it is easy to check that the weights of the vertices are distinct.this labeling construction shows that tvs(c n K m ) k. Combining this with the lower bound, we conclude that tvs(c n K m ) = k. Figure 7 shows the vertex irregular total labeling of C K Figure 7: tvs(c K ) = Theorem 2.8. tvs(p n K m ) = +nm 2,n,m 2. Proof. Let V(P n K m ) = v i,a j i : i n, j m} and E(P n K m ) = v i v i+,v i a j i : i n, j m}.let k = +nm 2, then from (4) it follows that,tvs(p n K m ) max +nm 2, +nm+n } n+ = +nm 2 = k. That is tvs(pn K m ) k.toprovethereverseinequality,wedefineafunctionf fromv E to,2,,...,k} in the following way k, if i = f(v i ) = i, if 2 i n k, if i = n; f(v i v i+ ) = k, i n; f(a i) = i,i i n;

13 9 Jeyanthi / Journal of Algorithms and Computation 48 (206) PP f(v i a j i ) = min+(n )(j ),k}; i n, j m. For i n,2 j m,+(n )(j ) < k f(a j i ) = i+j. For i n,2 j m,+(n )(j ) k f(a j i ) = (j )n+ k +i. We observe that for i n, j m, wt(a j i ) = (j )n++i. Also the weights of v i s are distinct.it is easy to check that the weights of the vertices are distinct.this labeling construction shows that tvs(p n K m ) k. Combining this with the lower bound, we conclude that tvs(p n K m ) = k. Figure 8 shows the vertex irregular total labeling of P K Figure 8: tvs(p K 4 ) = References [] [] A. Ahmad, K. M. Awan, I.Javaid, and Slamin, Total vertex irregularity strength of wheel related graphs, Australas.J.Combin., (20),47-6. [2] A. Ahmad, M.Bača and M. Numan, On irregularity strength of disjoint union of friendship graphs, Elect. J. Graph Th. App., (2) (20), [] A. Ahmad, S.A. Bokhary, M.Imran, A.Q. Baig, Total vertex irregularity strength of cubic graphs, Utilitas Math., 9: (20), [4] Ashfaq Ahmad, Syed Ahtsham ul Haq Bokhary, Roslan Hasni and Slamin, Total vertex irregularity strength of ladder related graphs, Sci.Int,26(), (204), -. [] M. Bača,S. Jendroľ, M. Miller and J. Ryan, On irregular total labellings, Discrete Math.,07 (2007), [6] T. Bohman and D. Kravitz, On the irregularity strength of trees, J. Graph Theory, 4(2004),24-24.

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