ON THE BASIS NUMBER OF THE CORONA OF GRAPHS

The basis number b(G) of a graphG is defined to be the least integer k such thatG has a k-fold 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. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. 1