Decomposing toroidal graphs into circuits and edges

Erdős et al. (Canad. J. Math. 18 (1966) 106–112) conjecture that there exists a constant dce such that every simple graph on n vertices can be decomposed into at most dcen 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. © 2005 Elsevier B.V. All rights reserved.