Geodesic
The basis number of some special non-planar graphs
Czechoslovak Mathematical Journal, Tome 53 (2003) no. 2, pp. 225-240

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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 $m,n\ge 5$, the basis number $b(K_{m,n})$ of the complete bipartite graph $K_{m,n}$ is equal to 4 except for $K_{6,10}$, $K_{5,n}$ and $K_{6,n}$ with $n=5,6,7,8$. We determine the basis number of some particular non-planar graphs such as $K_{5,n}$ and $K_{6,n}$, $n=5,6,7,8$, and $r$-cages for $r=5,6,7,8$, and the Robertson graph.
Classification : 05C35, 05C38
Keywords: graphs; basis number; cycle space; basis 
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     author = {Alsardary, Salar Y. and Ali, Ali A.},
     title = {The basis number of some special non-planar graphs},
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     pages = {225--240},
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     zbl = {1021.05053},
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Alsardary, Salar Y.; Ali, Ali A. The basis number of some special non-planar graphs. Czechoslovak Mathematical Journal, Tome 53 (2003) no. 2, pp. 225-240. http://geodesic.mathdoc.fr/item/CMJ_2003__53_2_a0/