Kong. Res. J. 2(2) : 30-33, 2015 ISSN 2349-2694 Kongunadu Arts and Science College, Coimbatore.

b-CHROMATIC NUMBER OF CORONA PRODUCT OF CROWN GRAPH AND COMPLETE 311 WITH PATH GRAPH

Vijayalakshmi, D. and G. Mohanappriya Department of Mathematics, Kongunadu Arts & Science College, Coimbatore-641 029. E.mail: [email protected]; [email protected] ABSTRACT

DOI:10.26524/krj A b-coloring of a graph is a proper coloring where each color admits at least one node (called dominating node) adjacent to every other used color. The maximum number of colors needed to b-color a graph G is called the b-chromatic number and is denoted by φ(G). In this paper, we find the b-chromatic number and some of the structural properties of corona product of crown graph and with path graph. Keywords: Corona product, crown graph, complete bipartite graph, path graph.

1. INTRODUCTION V2 such that no edge has both endpoints in the same subset, and every possible edge that could connect A b-coloring by k-colors is a proper coloring vertices in different subsets is part of the graph. That of the vertices of graph G such that in each color is, it is a bipartite graph (V , V , E) such that for every classes there exists a that has neighbors in all 1 2 two vertices v ∈ V and v ∈ V , v v is an edge in E. A the other k-1 color classes. The b-chromatic number 1 1 2 2 1 2 complete bipartite graph with partitions of size φ(G) is the largest number k for which G admits a b- |V |=m and |V |=n, is denoted K (Balakrishnan, coloring with k-colors (Irving and Manlove, 1999). 1 2 m,n 2004; Balakrishnan and Ranganathan, 2012). The corona G1 ◦ G2 of two graphs G1 and G2 is defined as a graph obtained by taking one copy of G1 2.3. Fan Graph (which has p1 vertices) and p1 copies of G2 and attach one copy of G2 at every vertex of G1 (Harary, and A Fan graph Fm,n is defined as the graph join K , where K the empty graph on nodes is and 1972). m m Pn is the path on n nodes (Wikipedia). In this paper we find for which the largest 2.4. Path Graph number k for which corona product of crown graph and complete bipartite graph with path graph admits The path graph Pn is a tree with two nodes a b-coloring with k-colors. And also we find some of of vertex degree 1, and the other n-2 nodes of vertex its structural properties (Venkatachalam and degree (Harary, 1972). Vernold Vivin, 2010; Vernold Vivin and 2.5. Corona Product Venkatachalam, 2012; Vijayalakshmi and Thilagavathi, 2012) Corona product or simply corona of any graph G1 and graph G2, defined as the graph which is 2. Definition the disjoint union of one copy of G1 and |V1| copies 2.1. Crown Graph of G2 (|V1| is the number of vertices of G1) in which each vertex of the copy of G1 is connected to all A crown graph on 2n vertices is an vertices of a separate copy of G2 (Harary, 1972). undirected graph with two sets of vertices ui and vi and with an edge from ui to vj whenever i ≠ j. The 2.6. b-coloring crown graph can be viewed as a complete bipartite A b-coloring of a graph is a proper coloring graph from which the edges of a perfect matching such that every color class contains a vertex that is have been removed (Wikipedia). adjacent to all other color classes. The b-chromatic 2.2. Complete Bipartite Graph number of a graph G, denoted by φ (G), is the maximum number t such that G admits a b-coloring A complete bipartite graph is a graph whose with t colors (Irving and Manlove, 1999). vertices can be partitioned into two subsets V1 and

30 3. CORONA PRODUCT OF CROWN GRAPH WITH 3.2.1. Theorem PATH GRAPH 0 2 For any n ≥ 3, q[푆 푛 ◦푃푛 ] = 5푛 -3n 3.1. b-chromatic number of corona product of Crown 0 0 Graph with Path Graph Proof: q[(푆 푛 ◦푃푛 ] = Number of edges in 푆 + 푛2n x Number of edges in 퐹푛 3.1.1. Theorem 2 0 = 푛 − 푛 +2n (2n-1) For any n ≥ 3, φ[푆푛푛◦푃 ] = 2n. = 2 − + 4 2- 2n-1. Proof: Let 푆0 be any Crown graph with vertices, V = 푛 푛 푛 푛 , , , , 0 {푣1 ,푣2 ,…,푣푛 } and 푉 = {푣 ,푣 , … , 푣 } i.e. V(푆 ) = V U = 5푛2 − 3푛. 1 2 푛 푛 , 0 2 푉 . Let the edges of 퐿 푛 be E(푆 푛)={ 푒 푗 : 1 ≤ 푗 ≤ 푛 − 푛} where 푒 푗 is the edge connecting , 푣 푖 푎푛푑푣 푗 푓표푟푒푣푒푟푦푖 ≠ 푗 .

Let 푃푛 be ant path graph of length n-1 with 푙 n-vertices. V(푝 푛) ={푢 ij: 1 ≤ 푖 ≤ 2푛, 1 ≤ 푙 ≤ 푛, 1 ≤ 푗 ≤ 푛} and and 퐸 푃푛 be {epi ∶ 2n − 1 ≤ i ≤n-1}. By the definition of corona graph each 0 vertex in Sn is adjacent to every vertex copy of 푃푛 , 0 i.e. vertices of 푉(퐿 푛 ◦푃푛 ) = V(푆 푛) 푈 푉(푃푛 ) .Let 0 0 2 E[푆 푛 ◦푃푛 ] be E(푆 푛 ) U 퐸 푃푛 U{ei: 푛 − n + 1 ≤ i≤ 5푛2 −3n}. Consider the color class C = 3.2.1. Theorem {푐 1, 푐 2, , 푐 3, … , 푐 푛, 푐 푛+1, 푐 푛+2, … , 푐 2푛} to color the vertices of (푆 0 ◦푃푛 ). Assign the colors For any n ≥ 3, the vertex polynomial of 2 2 3 n+2 푐 1, 푐 , , 푐 , … , 푐 to푛 푣 1 ,푣 2 ,…,푣 푛, 푖 . 푒 . 푣 , ,푠 and (푆 0 ◦푃푛 ) be 4푛x + 2n − 4n x + 4x 2 3 푛 , , 푖 푛 푐 푛+1, 푐 푛+2, … , 푐 2푛 푡표 푣 푗 s , j = 1,2,3,…,n respectively ∆(G) k Proof: V((푆 0 ◦푃푛 ; x) = V x for every i≠ j, i,j= 1,2,…,n. 푛 푘 =1 k , , = No of vertices having degree 2× x2 + From the figure we see that, each 푣 푠 are 3 , , adjacent to every 푣 푗 푠 for every, , i not equal, , to j and No of vertices having degree 3× x + vice versa. Hence both 푣 푖 푠 and푣 푗 푠 earns its adjacent color for every i≠ j. To make the above No of vertices having degree n+2× xn+2 coloring to be b-chromatic proper coloring of V(푝 푙 ) 2 2 3 n+2 푛 by corresponding non-adjacent vertices of its 푣 푖 ’s or = 4푛x + 2n − 4n x + 2n x . 푣 , ′푠 respectively. Thus each color has the neighbour 3.2.3. Some Structural Properties of (푆 0 ◦ 푃푛 )n ≥ 3. 푛 in푗 the every other color class. Thus, φ[(푆 0 ◦푃푛 ] = 2n. 푛 Vertex 0 Propert Maxim Minim 0 Let us assume that φ[푆 0 푛◦푃 ]푛> 2n, let it be No. of No. of Polynomi ies um um al φ[푆 푛 ◦푃 ] = 2n+1. The graph [푆 ◦ Graphs Vertex Edges Degree Degree 푛 푛 푃 푛] must requires 2n+2 vertices of degree 2n+1, all with distinct color Path 2x+(n- n n-1 2 1 2 and each must have adjacent with all of the other Graph (푛2 )푥 Crown n-1 n-1 color class, but at least one color class which does 2n 2 푛 -n 0 Graph − 푛)푥 푛 −1 not have a color dominating vertex in [푆 푛 ◦푃푛 ], 4푛푥 2 which invalidates the definition of b-coloring. Hence, 5 2 − 2 0 0 2n(n+ 푛 φ[푆 푛 ◦푃푛 ] not e0qual to 2n+1, it must be less than (푃푆 푛 ◦ 3n 2n-1 2 −+ 4(2푛푛)푥 3 1) 2n+1 i.e. φ[푆 푛 ◦푃푛 ] = 2n . Thus, for any n ≥3, the b- 푛 ) chromatic number of corona graph of crown graph + (2푛)푥 푛+ with path graph is 2n. 4. CORONA PRODUCT OF COMPLETE BIPARTITE 3.2. Illustration: b-coloring of corona product of GRAPH WITH PATH GRAPH Crown Graph with Path Graph 4.1. b-chromatic number on Corona Product of Complete Bipartite Graph with Path Graph

31 4.1.1. Theorem vertex, which contradicts the definition of b-coloring. Hence, φ[퐾푚 ,푛 ◦푃푛 ] not equal to 2n+1, must be less For any n ≥ 3, φ ◦ 푃 푛 ] = 2n m = n than 2n+1 i.e. φ[퐾푚 ,푛 ◦푃푛 ] = 2n . Thus, for any n ≥3, [퐾푚,푛 2n+1 m > n the b-chromatic number of corona graph of complete 2n-1 m < n bipartite graph with path graph is 2n for each m = n. Proof: Let be any complete bipartite graph with 퐾푚,푛 , , , , vertices, V = {푣 1 ,푣 2 ,…,푣 푛 } and 푉 = {푣 ,푣 , … , 푣 } i.e. Case (ii) m>n 1 2 푛 , , V(퐾푚,푛 ) = V U 푉 . Let the edges 퐾푚,푛of be Consider the color class C2 = E(퐾푚,푛 )={푒 푗 : 1 ≤ 푗 ≤ 푛2} where 푒 푗 is the edge {c1, c2, , c3, … , cn , cn+1, cn+2, … , c2n+1} to color the , vertices of K(m,n ◦Pn), m < 푛. Assign the colors connecting 푣 푖 푎푛푑 푣 . to v ,v ,…,v , i. e. v, ,s and 푗 Let 푃 be an t path graph of length n-1 with c1 , c2, , c3, … , cn 1 2 n i 푛 , , 푙 cn+1, cn+2, … , c2n+1 tov j s , j = 1,2,3,…,m n-vertices. V(푝 푛) = {푢 ij: 1 ≤ 푖 ≤ 2푛, 1 ≤ 푙 ≤ 푛, 1 ≤ respectively. 푗 ≤ 푛} and 퐸 푃푛 be {epi ∶ 2n − 1 ≤ i ≤n-1} The remaining proof of the theorem follows By the definition of corona graph each immediately from case (i). Hence φ vertex in Km,n is adjacent to every vertex copy of 푃푛 , [ Km,n ◦Pn ] = 2n+1,m > n. i.e. vertices of 푉(퐾푚,푛◦푃푛 ) = V(퐾푚,푛) 푈푉(푃푛 ) . Case (iii) m n. n+1 n+2 2n−1 j s , j = 1,2,3,…,m respectively. Consider the color class C =

{푐1 , 푐2, , 푐3 , … , 푐푛 , 푐푛+1, 푐푛+2 , … , 푐2푛+1} to color the The remaining proof of the theorem follows vertices of (퐾푚 ,푛 ◦푃푛 ). The proof follows from the immediately from case (i). Hence φ following cases. [ Km,n ◦Pn ] = 2n-1, m < n. Hence theproof. Case (i) m=n 4.1.1. Illustration Corona Product of Complete Bipartite Graph with Path Graph Consider the color class 퐶, = 1 { 4.1.2. Theorem 푐1, 푐2, , 푐3, … , 푐푛 , 푐푛+1, 푐푛+2, … , 푐2푛 } to color the vertices of 퐾(푚 ,푛 ◦푃푛), 푚 = 푛. Assign the colors 2 For any m, n ≥ 3, q[( Km,n ◦ Pn )] = 2n +3mn-m-n. 푐 , 푐 , , 푐 , … , 푐 to 푣 ,푣 ,…,푣 , 푖. 푒.푣,푖 ,푠 and 1 2 3 푛 , , 1 2 푛 Proof: q[(S0◦P ] = Number of edges in K + 2n x n n 푐푛+1, 푐푛+2 , … , 푐푚 푡표푣푗 s , j = 1,2,3,…,m respectively. m,n , , Number of edges in Fan graph Fn From the figure we assure that, each 푣 푠 푖 , , = mn+ (m+n) (2n-1) are adjacent to every 푣푗 푠 for and vice versa. Hence both 푣, ,푠 and푣, ,푠 earns its adjacent color. To make = mn+2mn-m+2n2 − n . 푖 푗 the above coloring to be b-chromatic proper coloring = 2n2 + 3mn − m − n. 푙 of V(푝푛 ) by corresponding non-adjacent vertices of , its 푣푖 ’s or 푣푗 ′푠 respectively,and the remaining vertices are colored properly by the colors in the color class. Thus each color has the neighbor in the every other color class. Thus, φ[퐾푚 ,푛 ◦푃푛] = 2n.

Let us assume that φ[ (퐾푚 ,푛 ◦푃푛 ]> 2n, say φ[퐾푚 ,푛 ◦푃푛 ] = 2n+1. The graph [퐾푚 ,푛 ◦푃푛 ] must requires 2n+2 vertices of degree 2n+1, all with distinct color and each must have adjacent with all of the other color class which is not possible, since maximum degree of 퐾푚 ,푛 ◦푃푛 is 2n, hence at least one color class does not have the color dominating

32 4.1.3. Theorem For any m, n ≥ 3, the vertex polynomial of be 2 2 3 2푛 For any m, n ≥ 3, the vertex polynomial of ( Km,n ◦ Pn) 4푛푥 + (2푛 − 4푛)푥 + 2푛푥 , m > n. 2 2 3 2푛 ∆(G) be( Km,n ◦ Pn ) 4푛푥 + (2푛 − 4푛)푥 + 2푛푥 , m=n. Proof: V( K ◦ P ; x) = V xk m,n n 푘=1 k Proof: V( K ◦ P ; x) = ∆(G) V xk = No of vertices having degree 2× x2 + m,n n 푘=1 k = No of vertices having degree 2× x2 + No of vertices having degree 3× x3 +

No of vertices having degree 3× x3 + No of vertices having degree 2n-1× x2n+1 No of vertices having degree 2n× x2n + =(4푛 − 2)푥2 + (2푛2 − 3푛 − 2)푥3 + = 4푛x2 + 2n2 − 4n x3 + 2n xn+2. (2푛 + 1)푥2푛+1 . 4.1.4. Theorem

4.4.5. Some Structural Properties of ( 퐾푚,푛 ◦ 푃푛), n ≥ 3.

Properties Number of Number of Maximum Minimum Vertex Polynomial Graphs Vertex Edges Degree Degree Path Graph n n-1 2 1 2x+(n-2)푥2 2 Complete m+n 푛 max{m,n} min{m,n} (2푛)푥푛 Bipartite Graph 2 2 2 2 3 (퐾푚 ,푛 ◦ 푃푛) 2푛 + 2n 5푛 − 2n 2n 2 4푛푥 + (2푛 − 4푛)푥 2 m = n + 2푛푥 푛 2 2 2 2 (퐾푚 ,푛 ◦ 푃푛) 2푛 5푛 (4푛 − 2)푥 + (2푛 − 5푛 m < n +n-1 −5n+1 2n 2 + 2)푥3 + 푛푥2푛−1 + (푛 2 − 1)푥 푛 2 2 2 (퐾푚 ,푛 ◦ 푃푛) 2n 5푛 (4푛 + 2)푥 m > n +3n+1 +n-1 2n+1 2 +(2푛2 − 3푛 − 2)푥3 2 1 +(2푛 + 1)푥 푛+

5. CONCLUSION Irving, R.W. and D.F. Manlove, (1999). The b- chromatic number of a graph. Discrete. Appl. In this paper we operated the graph Math., 91: 127–141. operation corona product on crown graph and complete bipartite graph with path graph, we get Venkatachalam, M. and J. Vernold Vivin, (2010). The corona product of crown graph with path graph and b-chromatic number of star graph families. Le- corona product of complete bipartite graph with Matematica, 65(1): 119–125. path graph and also we find its b-chromatic number Vernold Vivin, J. and M. Venkatachalam, (2012). The and some of its structural properties. b-Chromatic Number of Corona Graphs, Utilitas REFERENCES Mathematica, 88: 299-307. Balakrishnan, R. and K. Ranganathan, (2012). A Vijayalakshmi, D. and K. Thilagavathi, (2012). Study textbook of . Springer, New York. On b-Chromatic Coloring Of Some Graphs. Ph.D., Thesis, Bharathiar University. Balakrishnan, V.K. (2004). Graph Theory. Tata McGraw-Hill Publishing Ltd, New Delhi. Wikipedia, the free Encyclopedia. Harary, F. (1972). Graph Theory. Addison-Wesley, Reading, Massachusetts.

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