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The basis number of the join of graphs. (English) Zbl 0806.05043

Summary: The basis number of a graph \(G\) is defined to be the least integer \(h\) such that \(G\) has an \(h\)-fold basis for its cycle space. MacLane has shown that a graph is planar if and only if its basis number is not greater than 2. The basis numbers of the complete graphs, complete bipartite graphs, and the \(n\)-cube have been determined by Schmeichel and Banks, see E. F. Schmeichel [J. Comb. Theory, Ser. B 30, No. 2, 123-129 (1981; Zbl 0385.05031)] and J. A. Banks and E. F. Schmeichel [J. Comb. Theory, Ser. B 33, No. 2, 95-100 (1982; Zbl 0502.05054)]. We investigate the basis number of the join of two graphs.

MSC:

05C38 Paths and cycles
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