This thesis is divided into two parts. In the first part a game or a process on a graph, called chip-firing game has been studied. The chip-firing game is one person game or a diffusion process on a graph. Although it has been around for no more than 20 years, but it has rapidly become an important and interesting object of study in structural combinatorics. The reason for this is due to its relation with important concepts like random walks, Laplacian matrix, the Tutte polynomial, the chromatic polynomial, algebraic graph theory, group theory and theoretical physics. So far many variants of the chip-firing game have been defined and studied. In this part we survey results on the chip-firing game on graphs, directed graphs, and also a variant of it called dollar game and overview the numerous connections that the chip-firing game has with some other parts of combinatorics. In the second part we answer to some open problems on a kind of vertex coloring on graphs called b-coloring. In this part the b-coloring of cartesian products of paths, cycles and complete graphs, also Kneser graphs has been studied.