Recurrcnce properties for random walks on a two-dimensional random graph. In [1] a random walk on a bi-dimensional random graph was studied. This model had been previously introduced in the physical literature. and studied numerically. It is established in [1] that, in contrast with the corresponding periodic graph, this random walk is transient with probability one with respect to the random environment. Subsequently several papers appeared on related models ([3], [4] [5] [6]). The results of [1] have been used in [7] to study a model of corner percolation on Z2. In [2] transience was obtained on a more general random environment and a transition from recurrence to transience has been proved to occur. At the moment work is going on in collaboration with G. Bosi to study the problem on a honeycomb random graph. [1] M. Campanino, D. Petritis. Random walks on randomly oriented lattices. Markov Process. Relat. Fields 9, 391-412 (2003). [2] M. Campanino, D. Petritis. Type transition of simple random walks on randomly directed regular lattices, J. Appl. Prob. 51, 1065-1080 (2014). [3] B. De Loynes. Marche aleatoire sur un di-graphe et frontière de Martin. C. R. Acad. Sci. Paris 350, 87-90 (2012). [4] A. Devuldier, F. Pene. Random walk in random environment in a two-dimensional stratified medium with orientation. Electron. J. Prob. 18, no. 18 (2013). [5] Guillotin-Plantard, A, Le Ny. A functional limit theorem for a 2D-random walk with dependent marginals. Electron. Commun. Prob. 13, 337-351 (2008). [6] Guillotin-Plantard, A, Le Ny. Transient random walks on 2D-oriented lattice. Theory Prob. Appl. 52, 699-711 (2008). [7] G. Pete. Corner percolation on Z2 and the square root of 17. Ann. Prob. 36, 1711-1747 (2008).