Abstract
Erdős et al. (Canad. J. Math. 18 (1966) 106–112) conjecture that there exists a constant d ce such that every simple graph on n vertices can be decomposed into at most d ce n circuits and edges. We consider toroidal graphs, where the graphs can be embedded on the torus, and give a polynomial time algorithm to decompose the edge set of an even toroidal graph on n vertices into at most ( n + 3 ) / 2 circuits. As a corollary, we get a polynomial time algorithm to decompose the edge set of a toroidal graph (not necessarily even) on n vertices into at most 3 ( n - 1 ) / 2 circuits and edges. This settles the conjecture for toroidal graphs.
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