On the basis number of the corona of graphs

Abstract

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∘T)=2 for any tree and any vertex v not in T, 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.