We study analytical and numerical properties of the L 1 - L 2 minimization problem for sparse representation of a signal over a highly coherent dictionary. Though the L 1 - L 2 metric is non-convex, it is Lipschitz continuous. The difference of convex algorithm (DCA) is readily applicable for computing the sparse representation coefficients. The L 1 - L 2 minimization appears as an initialization step of DCA. We further integrate DCA with a non-standard simulated annealing methodology to approximate globally sparse solutions. Non-Gaussian random perturbations are more effective than standard Gaussian perturbations for improving sparsity of solutions. In numerical experiments, we conduct an extensive comparison among sparse penalties such as L 1 - L 2 for L 1 - L 2 based on data from three specific applications (over-sampled discreet cosine basis, differential absorption optical spectroscopy, and image denoising) where

Let $f(n)$ be the maximum number of edges in a graph on $n$ vertices in which no two cycles have the same length. P. Erdos raised the problem of determining $f(n)$ (see J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, New York, 1976), p.247, Problem 11). We present the problems, conjectures related to this problems and we summarize the know results. We make the following conjecture: \par \noindent{\bf Conjecture } $$\lim\sb {n \to \infty} {f(n)-n \over \sqrt n} = \sqrt {2 + \frac{4}{9}}.$$

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