ON THE BASIS NUMBER OF THE CORONA OF GRAPHS

The basis number b(G) of a graph G is defined to be the least integer k such that G has a kfold basis for its cycle space. In this note, we determine the basis number of the corona of graphs, in fact we prove that b(v ◦T)= 2 for any tree and any vertex v not inT , b(v ◦H)≤ b(H) + 2, where H is any graph and v is not a vertex of H , also we prove that if G= G1 ◦ G2 is the corona of two graphs G1 and G2, then b(G1) ≤ b(G) ≤max{b(G1),b(G2) + 2}, moreover we prove that if G is a Hamiltonian graph, then b(v ◦G) ≤ b(G) + 1, where v is any vertex not in G, and finally we give a sequence of remarks which gives the basis number of the corona of some of special graphs.


Introduction
In this note, we consider only finite, undirected, simple graphs.Our terminology and notation will be standard except as indicated.For undefined terms, see [7].Let G be a (p, q) graph (i.e., G has p vertices and q edges), and let e 1 ,e 2 ,...,e q be an ordering of its edges.Then any subset E of edges in G corresponds to (0,1)-vector (v 1 ,...,v q ) with v i = 1 if e i ∈ E and v i = 0 if e i / ∈ E. The vectors form a q-dimensional vector space over the field of two elements Z 2 and is denoted by (Z 2 ) q .The vectors in (Z 2 ) q which correspond to the cycles in G generate a subspace called the cycle space of G and is denoted by C(G), we will say that the cycles themselves, instead of saying the vectors corresponding to the cycles, generate C(G).It is well known (see [7, page 39]) that if G is a (p, q) graph with k components, then dimC(G) = γ(G) = q − p + k, where γ(G) is the cyclomatic number of G.A basis for C(G) is called a k-fold basis if each edge of G occurs in at most k of the cycles in the basis.The basis number of G denoted by b(G) is the smallest integer k such that C(G) has a k-fold basis.The corona (see [7, page 167]) of two graphs G 1 and G 2 , denoted by G 1 • G 2 , is defined to be the graph G obtained by taking one copy of G 1 (which has p 1 vertices) and p 1 copies of G 2 , and then joining the vertex of G 1 to every vertex in the ith copy of G 2 .If G 1 is a (p 1 , q 1 ) graph and G 2 is a (p 2 , q 2 ) graph, then it follows from the definition of the corona that G 1 • G 2 has p 1 (1 + p 2 ) vertices and q 1 + p 1 q 2 + p (see [7, page 168] In the rest of this note, P n , C n , S n , and W n stand for the path, the cycle, the star, and the wheel of n vertices.A theta graph θ n is defined to be a cycle C n with n vertices, respectively, to which we add a new edge that joins two nonadjacent vertices of C n . MacLane [8] proved that a graph G is planar if and only if b(G) ≤ 2. Schmeichel [9] proved that for n ≥ 5, b(K n ) = 3 and for m,n ≥ 5, b(K m,n ) = 4 except for K 6,10 , K 5,n , and K 6,n , where n = 5, 6, 7, and 8. Banks and Schmeichel [6] proved that for n The purpose of this note is to investigate the basis number of the corona of graphs, in fact we prove that for any two graphs ,b(G 2 ) + 2} and we give the exact basis number of the corona of some special graphs.

Main results
This section is devoted for proving the main results of this note, and this is done by writing a sequence of theorems and remarks.
where G is any one of the following graphs: Lemma 2.2.Let T be a tree with p vertices (p ≥ 3) if v is any point which is not a vertex of T, and if G = v • T, then b(G) = 2, and hence G is planar.
Proof.Assume that G is not planar.Then, by Kuratowski's theorem, G contains a subdivision of K 5 or K 3,3 .Then G − x cannot be acyclic graph for any x ∈ V (G), while G − v is a tree.This is a contradiction, and hence Lemma 2.3.Let H be any connected (p, q) graph and let v be any vertex which is not a vertex of H.
Proof.Let u 1 ,u 2 ,...,u p be the vertices of H. Since dimC(G) = q and dimC(H) = q − p + 1, dimC(G) − dim C(H) = p − 1.Let T be a spanning tree of H. Then b(v • T) = 2, dimC(v • T) = p − 1, and each cycle in v • T must contain an edge of the form vu i for some i ∈ {1, 2,..., p}.Thus the cycles in v • T are independent from the cycles in H. Let B 1 be a b(H)-fold basis for C(H), and let B 2 be a 2-fold basis for v • T. Then clearly

1 p 2 edges
Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 53712, Pages 1-3 DOI 10.1155/IJMMS/2006/53712 and n is even where ∧ and × are the direct and the cartesian products of graphs, respectively.Next we restate [3, Theorem 2.3].Theorem 1.1.Let G be a graph obtained from G by deleting an edge e of at most 2-fold in a basis B for C