The Achromatic and b- Chromatic Colouring of Central Graph of Book Graph and Shadow graph of Path graph

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1 Volume No. 0, 9 ISSN: -00 (printed version); ISSN: -9 (on-line version) url: ijpam.eu The Achromatic and b- Chromatic Colouring of Central Graph of Book Graph and Shadow graph of Path graph K.P.Thilagavathy anda.santha Department of Mathematics, Kumaraguru College of Technology, Coimbatore, India. kp Santha jsk@yahoo.co.in February, 0 Abstract Irving and Manlove introduced a b-chromatic colouring which is a refinement of an achromatic colouring. They studied the properties of the b-chromatic number.they proved that it is NP- complete. Also they proved that the b- chromatic number is polynomial time computable for tree. In this research work the achromatic and b-chromatic number of the central graph of shadow graph of path graph are found. Also the achromatic an b-chromatic numbers of central graphs of some graphs namely triangular book graph, quadrilateral book graph, bull graph and Petersen graphs are studied. AMS Subject Classification: 0C Key Words and Phrases:Achromatic Number, b-chromatic number, Book graph, Shadow graph, stacked book graph Introduction In this paper, by a graph we mean a finite undirected, simple graph. For a given graph G = (V, E) an operation is done on G, by ijpam.eu 0

2 dividing each edge of G once and joining all the non- adjacent vertices of G. The graph obtained by this method is called central graph of G and it is denoted by C(G).The Shadow graph D (G) of a connected graph G is constructed by taking two copies of G say G and G. Join each vertex u in G to the neighbours of the corresponding vertex v in G. The m book graph is defined as the graph Cartesian product S m+ P where S m is the Star graph and P is the path graph on two nodes.a triangular book BK n is the complete tripartite graph consisting of K,,n.It is graph consisting of n triangles sharing a common edge. Bull graph Bl is a planar undirected graph with vertices and edges in the form of a triangle with two disjoint pendent edges. An achromatic colouring is a proper vertex colouring such that each pair of colours is adjacent by at least one edge. The largest possible number of colours in an achromatic colouring of G is called the achromatic number and it is denoted by ψ(g). The b-chromatic number ϕ(g)[] of a graph G is the largest integer k, such that G admits a proper k colouring, and every colour class has a representative vertex adjacent at least to one vertex in each other class. Theorem. For any path graph P n,ψ[c(d (P n ))] = n, n > Proof. Let D (P n )be a shadow graph of P n which is obtained by taking two copies of P n say P n and P. n Join each vertex u of P nto the neighbours of the corresponding vertex v of P n. Consider the two vertex sets U = {u, u,..., u n }and V = {v, v,..., v n }. Assign the path graph P n by the vertices in U and P n by the vertices in V. In C(D (P n )), let v i,i+ and u i,i+ be the newly introduced vertex on the edge joining v i v i+ and u i u i+ respectively where i =,,, n. Also let v i u i+,u i v i+ be the newly introduced vertex on the edge joining v i u i+ and u i v i+ respectively where i =,,, n. Consider the two set of colours C = {C, C,..., C n }and C = {C, C,..., C n}. For i n assign C i to u i and C i to v i.to make the colouring as achromatic consider the following assignment of colours: For i n, assign C i+ to the vertex v i u i+ ijpam.eu 0

3 For i n, assign C i+to the vertex u i v i+ For i n, assign C i+ to the vertex v i v i+ For i n, assign C i+ to the vertex u i,i+ If we introduce a new C n+ to any vertex in the graph, that will not adjacent to all the other colours in the colour set. Hence the maximum possible colouring is C C. That is ψ[c(d (P n ))] = n ψ[c(d (P ))] = 0 Figure- { Corollary. For any path graph P n, ϕ[c(d (P n ))] = n + n, n + n, n = even n = odd The Achromatic and b-chromatic number of Book Graphs Theorem. For a triangular book graph BK n, ψ[c(bk n )] = n + ijpam.eu 0

4 Proof. Let BK n be a book graph with n pages consisting of n triangles sharing one edge. Consider the vertex set V = {v 0, v,.., v n, v n+ }. Let v 0, v n+ be the vertices of the path graph. And let v,.., v n be the vertices of the n pages of the graph BK n in the anti - clock wise direction. In C(BK n ), let v 0,n+ be the newly introduced vertex on the edge joining v 0, v n+. Let v 0,i be the vertex on the edge joining v 0, v i where i =,,,..., n and let v n+,i be the vertex joining the edge v n+, v i,i =,,,..., n. Consider the colour set C = {C 0, C,..., C n+ }. For 0 i n +, assign C i to V i. To make the colouring as achromatic consider the following colouring procedure: For i n, assign the colour C i to the vertex v n+,i. For i n, assign the colour C i to the vertex v 0,i Assign a colour C n to the vertex v 0,n+ and v 0,. If we introduce any new colour say C n+ to any vertex that will not be adjacent to all the other colours in the colour set. Hence by this assignment of colours it is the maximal one and hence it is achromatic. ψ[c(bk )] = Figure- Theorem. For a triangular book graph BK n, ϕ[c(bk n ))] = n + ijpam.eu 0

5 Proof. Let BK n be a book graph with n pages consisting of n triangles sharing one edge. Consider the vertex set U = {u 0, u,.., u n, u n+ }and u 0 and u n+ be the vertices of the path graph and let u, u,.., u n be the vertices of the n pages of the graph BK n. Name the vertices of the book graph in the anticlock wise direction. In C(BK n ) let u 0,n+ be the newly introduced vertex joining u 0 u n+. For i n, let u 0,i and u i,n+ be the newly introduced vertex joining u 0 u i and u i u n+ respectively. Consider the set of colours C = {C 0, C,, C n }.For i n assign C i to u i also assign C n to u n+. To make the colouring as b-chromatic consider the following procedure: Assign a colour C to the vertex u 0,n+. For i n, assign C 0 to the vertex u i,n+. For i n, assign C i to the vertex u 0,i. Assign a colour C n to the vertex u 0,. By this assignment of colours it is the maximum and it is b-chromatic. Theorem. For any quadrilateral book graph QB n, ψ[c(qb n )] = n + Proof. Let QB n be the quadrilateral book graph of order (, n) is defined as the graph Cartesian product of the star graph S n+ and the path graph P on vertices. Consider the two vertex sets U and V where U = {u 0, u,.., u n }and V = {v 0, v,.., v n }. Assign u 0 and v 0 to the path vertices and assign u,.., u n and v,.., v n in the anti clock wise direction in such a way that u i and v i for i =,,.., n are in the same page and are adjacent to each other and to u 0 and v 0 respectively. In C(QB n ), let the newly introduced vertex in the edge joining u 0 v 0 be uv 0, and the vertex in the edge u 0 -u i be u 0,i.Also let uv i be the newly introduced vertex on the edge joining u i v i and v 0,i be the vertex on the edge joining v 0 v i where i =,,,.., n. Consider the two set of colours C = {C 0, C,.., C n }and C = {C 0, C,.., C n}. For 0 i n, assign C i to u i and C ito v i. To make the colouring as achromatic consider the following colouring procedure: ijpam.eu 0

6 For i n, assign C 0 to the vertex uv i. For i n, assign C i to the vertex v 0,i. For i n, assign C i to the vertex u 0,i. Assign C nto the vertex uv 0 and assign a colour C 0 to the vertex v o,. If we introduce any new colour C n+ to any vertex that will not adjacent to all the remaining colours. Hence the maximum possible colouring is C C. Hence ψ[c(qb n )] = n +. ψ[c(qb )] =. Figure- Corollary. For any quadrilateral book graph QBn,ϕ[C(QB n )] = n + Observations For the Bull graph Bl, ψ[c(d (Bl))] = 0 For the Bull graph Bl, ϕ[c(d (Bl))] = For the Petersen graph, ψ[c(p etersen Graph))] = 0 ijpam.eu 0

7 0 0 9 ψ[c(p etersen Graph))] = 0 Figure- For the Petersen graph, ϕ[c(p etersen Graph))] = References [] Irving, R.W and Manlove, D.F, 999, The b-chromatic number of a graph, Discrete Applied Mathematics 9(-) -. [] Jonathan gross, Jay yellan, 00 Handbook of graph theory CRC press, New York,. [] Thilagavathy, K.P, Santha. A, 0 A Note on Achromatic and b-chromatic number of graphs International Journal of Applied Mathematics and Statistics,(),0-0. [] Vaidya S.K, Shah N.H, 0, Some new odd harmonious graphs International Journal of Mathematics and Soft Computing Vol., No. (0), 9 -. ijpam.eu 0

8 [] Vernold Vivin, J. Thilagavathi, K, 00, On Harmonious colouring of Central Graphs, Far East. Math. Sci (FJMS),,9-9. [] West, D. B. Introduction to Graph Theory, nd ed. Englewood Cliffs, NJ: Prentice-Hall, p., 000. ijpam.eu 0

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