The Basis Number of Some Special Non-Planar Graphs

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The basis number of a graph G was defined by Schmeichel to be the least integer h such that G has an h-fold basis for its cycle space. He proved that for m, n ≥ 5, the basis number b(Km,n) of the complete bipartite graph Km,n is equal to 4 except for K6,10, K5,n and K6,n with n = 5, 6, 7, 8. We determine the basis number of some particular non-planar graphs such as K5,n and K6,n, n = 5, 6, 7, 8, and r-cages for r = 5, 6, 7, 8, and the Robertson graph.

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