SciresolSciresolhttps://indjst.org/author-guidelinesIndian Journal of Science and Technology0974-564510.17485/IJST/v16i9.1910Research ArticleProper d-Lucky Labelingof Corona Products of Certain GraphsKujurChiranjilalchiranji@yahoo.com1Assistant Professor, St. Joseph's CollegeDarjeeling, West Bengal, 734426India832023169668219202241120222023Abstract

Objectives:The objective of this paper is to find exact proper d-lucky number for corona product graphs of certain graphs. Methods:The results are proved by method of elaboration and construction. Findings:Exact proper d-lucky numbers are obtained for corona products: Pn⊙Cm, Pn⊙Km, Cn⊙Kn and Kn⊙Kn. Novelty: Proper d-lucky labeling for the corona product of Pn⊙Cm, Pn⊙Km, Cn⊙Kn and Kn⊙Knare not studied by any other authors, thus the proper d-lucky numbers for the above graphs are new findings.

A lot of research work is under progress in the area of graph labeling and in its variants. Lucky and Proper Lucky Labeling of Quadrilateral Snake Graphs were studied by T. V. Sateesh Kumar et al 1. d- Lucky labeling was introduced by Mirka Miller et al 2. A lower bound and several exact results on d-lucky number were found by Sandi Klavzar et al 3. Some work on d-lucky labeling of Honeycomb network were done by A. Sahayamary et al 4. Cordial labeling of certain corona product graphs was obtained by Elrokh, A. et al 5. In this paper proper d-lucky number is defined and new notation for the same is introduced and proper d-lucky number for corona products of certain graphs are obtained.

Proper d-lucky labeling for Corona Product of <inline-formula id="if-5c336f02363e"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="bold-italic">P</mml:mi><mml:mi mathvariant="bold-italic">n</mml:mi></mml:msub><mml:mo>⊙</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">C</mml:mi><mml:mi mathvariant="bold-italic">m</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">P</mml:mi><mml:mi mathvariant="bold-italic">n</mml:mi></mml:msub><mml:mo>⊙</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">K</mml:mi><mml:mi mathvariant="bold-italic">m</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">C</mml:mi><mml:mi mathvariant="bold-italic">n</mml:mi></mml:msub><mml:mo>⊙</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">K</mml:mi><mml:mi mathvariant="bold-italic">n</mml:mi></mml:msub><mml:mtext> and </mml:mtext><mml:msub><mml:mi mathvariant="bold-italic">K</mml:mi><mml:mi mathvariant="bold-italic">n</mml:mi></mml:msub><mml:mo>⊙</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">K</mml:mi><mml:mi mathvariant="bold-italic">n</mml:mi></mml:msub></mml:math></inline-formula>

For a vertex u in a graph G, let Nu={v∈V(G)/uv∈E(G). Let l:VG→{1,2,…,k} be a labeling of vertices of a graph G by positive integers. Define Cu=∑v∈N(u)lv+d(u), where d(u) denotes the degree ofu. Define a labeling l as d-lucky if Cu≠C(v), for every pair of adjacent vertices u and v in G. The d-lucky number of a graph G, denoted byɳdl(G), is the least positive k such that G has a d-lucky labeling with {1,2,…,k} as the set of labels 2. A d-lucky labeling is said to be proper d-lucky labeling if for every pair of adjacent vertices uandv in G, u≠v, and is denoted by ƞpdl(G).

The corona product of G and H is the graph G⊙H obtained by taking one copy of G, called the center graph, V(G) copies of H, called the outer graph, and making the ith vertex of G adjacent to every vertex of the ith copy of H, where 1≤i≤V(G). We take first corona product of path Pn with cycle graph Cn and Pn with complete graph Kn as Cn⊙Kn and Kn⊙Kn then compute the proper d-lucky labeling for them and obtain the proper d-lucky number for the same.

Theorem 1:

Corona product of Pn⊙Cm admits proper d-lucky labeling and ηpdlPn⊙Cm=3, when m is even 5, when m is odd .

Proof:

The theorem is proved by construction. There are two cases.

Case 1: When m is even

Label the base vertices of path Pn with 1,2 alternately.

If vertices of Cm is adjacent to a vertex which has label as 1 in the path, then, label all other vertices of Cn with 2,3 alternately, if it is adjacent to vertex with label as 2 then label all other vertices of Cn with 1,3 alternately.

The neighborhood sums su’s and cu's of Cm are calculated as follows:

The vertices with label as 2 have su=7 and cu=10. The vertices with label as 3 which are adjacent to path vertex with label as 1 then, su=5 and cu=8. The vertices with label as 1 have su=8 and cu=11. The vertices with label as 3 which are adjacent to vertex with label as 2 in the path have su=4 and cu=7.

In path Pn the vertices with label as 1 have su=5m+82 and cu=5m+202 except the end vertices have, su=5m+42 and cu=7m+62. In path the vertices with label as 2 have

su=2m+2 and cu=3m+4, except the end vertex has su=2m+1 and cu=3m+2.

Case 2: When m is odd

Label the vertices of base path Pn with 4,5 alternately. Label the vertices of Cm if they are adjacent to the path vertex labeled as 4 then, with 1,3 alternately and last vertex with 2. If they are adjacent to path vertex having label as 5 then label all other vertices as 1,3 alternately and the last vertex as 2. (for illustration see Figure 1)

The neighborhood sums s(u)'s and c(u)'s for Cm are calculated as follows:

The vertices which are adjacent to path vertex labeled as 4 or labeled as 1 have su=9or10 and cu=12or13. The vertices labeled as 3 have su=6or7andcu=9or10. The vertices labeled as 2 have su=8andcu=11.

If the vertices of Cm are adjacent to path vertex labeled as 5 then, the vertices in Cn labeled as 1 have su=10or11 and cu=13or14. The vertices labeled as 3 have su=7or8 and cu=10or11. the vertices labeled as 2 have su=9 and cu=11.

In the path, vertices labeled as 4 have su=2m+10 and cu=3m+12, except the end vertex which has su=2m+5 and cu=3m+6.

The vertices labeled as 5 in the path have su=2m+8 and cu=3m+10, except the end vertex which has su=2m+4 and cu=3m+5.

It is observed that no two adjacent vertices have the same c(u)'s .

Hence Corona product of Pn⊙Cm admits proper d-lucky labeling and -

Theorem 2:

Corona product of Pn⊙Km admits proper d-lucky labeling and ɳpdlPn⊙Km=m+1

Proof:

Label the vertices of the base path with 1,2,3 in cyclic order. Vertices of Km if they are adjacent to path vertex with label as 1 then label all other vertices of Km with 2,3,4,…,(m+1). If they are adjacent to vertex with label as 2 then label them with 3,4,5,…,m+1,1, and if they are adjacent to vertex with label as 3 then label them with 4,5,6,…,m+1,2,1 etc.

2 proper d-lucky labeling of <inline-formula id="if-c5eec9c07f15"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>P</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>⊙</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mn>5</mml:mn></mml:msub></mml:math></inline-formula>

The neighborhood sums su's and c(u)'s for Km are calculated as follows:

fori=1,2,3,…,,m+1sui=m+1m+2-2i2cui=m2+5m-2(i-1)2

The neighborhood sums s(u)'s and c(u)'s for path vertices are calculated as follows:

The vertices with label as 1 have su=m+1m+2+82

cu=m2+5m+142, except the end vertex which is adjacent to 2 has su=m+1m+2+22,

cu=m2+5m+62 and if the vertex is adjacent to 3 then su=m+1m+2+42 and

cu=m2+5m+82 .

The path vertices with label as 2 have su=m+1m+2+42

and cu=m2+5m+102 except the end vertex which has su=m+1m+2-22 and cu=m2+5m+22.

The path vertex with label as 3 has u=(m+1)(m+2)2,cu=m2+5m+62 , except the end vertex , which has su=m+1m+2-22andcu=m2+5m+22.

It is seen that no two adjacent vertices have the same c(u)'s.

Hence Corona product of Pn⊙Km admits proper d-lucky labeling and ɳpdlPn⊙Km=m+1.

Theorem 3:

The corona product of Cn⊙Kn admits proper d-lucky labeling and ɳpdlCn⊙Kn=n+1.

Proof: label the base vertices of Cn with 1,2,3,…,n.

Label the vertices of Kn which are adjacent to vertex with labeled as 1 of Cn as 2,3,4,…,(n+1). If the vertices are adjacent to vertex with label as 2 of base Cn then label the vertices of Kn as 3,4,5,…,(n+1),1 and so on till all the vertices of Kn's are labelled.

Proper d-lucky labeling of <inline-formula id="if-1ef08b56b6a3"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>C</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>⊙</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula>

The neighborhood sums s(u)'s and c(u)'s of Kn's are calculated as follows:

fori=1,2,3,…(n+1)

sui=n+1n+2-2i2 and cui=n2+5n-2(i-1)2.

The neighborhood sums s(u)'s and c(u)'s for the base vertices of Cn's are calculated as follows:

The base vertices with label as 1 have su=n2+5n+42 and cu=n2+7n+82 and the vertices with label as n have, su=(n+1)(n+2)2 and cu=n2+5n+62.

For all other vertices of Cnfori=2,3,4,…(n-1)

sui=n+1n+2+2i2 and cui=n2+5n+2(3+i)2.

It is observed that no two adjacent vertices have the same c(u)'s.

Hence the corona product of Cn⊙Kn admits proper d-lucky labeling and ɳpdlCn⊙Kn=n+1.

Theorem 4:

The corona product of Kn⊙Kn admits proper d-lucky labeling and ɳpdlKn⊙Kn=n+1.

Proof: Label the base vertices of Kn with 1,2,3,…,n.

The outer Kn vertices if they are adjacent to vertex with label as 1 then label them as 2,3,4,…,(n+1). If they are adjacent to vertex with labeled as 2 then label them as 3,4,5,…,(n+1),1 and so on till all the outer vertices of Kn's are labeled.

Proper d-lucky labeling of <inline-formula id="if-0f1754b6479d"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>K</mml:mi><mml:mn>5</mml:mn></mml:msub><mml:mo>⊙</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mn>5</mml:mn></mml:msub></mml:math></inline-formula>

The neighborhood sums s(u) and c(u)'s for outer Kn's are calculated as follows:

fori=1,2,3,…,(n+1)sui=n+1n+2-2i2cui=n2+5n-2(i-1)2

For base Kn vertices the neighborhood sums s(u) and c(u)'s are given by

fori=1,2,3,…,n.sui=(n+1)2-2icui=n2+4n-2i

It is observed that no two adjacent vertices have the same c(u)'s.

Hence the corona product of Kn⊙Kn admits proper d-lucky labeling and

ɳpdlKn⊙Kn=n+1.

Conclusion

In this paper proper d-lucky number for corona product graphs of Pn⊙Cm, Pn⊙Km, Cn⊙Kn and Kn⊙Knare computed and found as ηpdlPn⊙Cm=3, when m is even 5, when m is odd ,

I acknowledge the valuable suggestions of the reviewers and thank them profoundly for their contributions they made, for the improvement of my article.

ReferencesKumar T V SateeshMeenakshiSLucky and Proper Lucky Labeling of Quadrilateral Snake GraphsMillerMirkaRajasinghIndraD Ahima EmiletJemiletD Azubhad-Lucky Labeling of GraphsKlavžarSandiRajasinghIndraEmiletD AhimaA lower bound and several exact results on the d-lucky numberARini Angeline SahayamarySTeresa Arockiamaryd-lucky labeling of Honeycomb NetworkElrokhAshraf Ibrahim HefnawyNadaShokry Ibrahim MohamedEl-ShafeyEman Mohamed El-SayedCordial Labeling of Corona Product of Path Graph and Second Power of Fan Graph