On the basis number of the corona of graphs (Q885626)

Summary: The basis number \(b(G)\) of a graph \(G\) is defined to be the least integer \(k\) such that \(G\) 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 \(b(v\circ T)=2\) for any tree and any vertex \(v\) not in \(T\), \(b(v\circ H)\leq b(H)+2\), where \(H\) is any graph and \(v\) is not a vertex of \(H\), also we prove that if \(G= G_1\circ G_2\) is the corona of two graphs \(G_1\) and \(G_2\), then \(b(G_1)\leq b(G)\leq \max\{b(G_1),b(G_2)+2\}\), moreover we prove that if \(G\) is a Hamiltonian graph, then \(b(v\circ G)\leq b(G)+1\), 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 the special graphs.