On the basis number of ternary join of graphs

Abstract EN

The basis number, b(G), of a graph G is defined to be the smallest positive integer k such that G has a k-fold basis for its cycle space.

We investigate an upper bound for b(G]+G2+G3).

It is proved that, if Gr G2 and G3 are vertex-disjoint graphs, and each has a spanning tree of valency not more than 4, then b(G1+G2+G3)< max {4, b (G^+l, b(G2)+2, b(G3)+l}.

The basis number of ternary join of paths, cycles, wheels and complement of complete graphs is discussed and it is proved that b(P +P +P )= b (C + C + C ) 4, for n,m,p ≥ 9, v n, m, p≥12 b(Wn+Wm+Wp) = 4, for n,m,p >12, b( K n+ K m + Kp ) = 4, for n,m,p ≥6.