| This thesis contains three parts. The first part characterizes the retracts of Cartesian product graphs with one factor being strongly (2k + 1)-angulated or weakly (2k + 1)-angulated. In the second part, we apply the results to construct cores of Cartesian product form using vertex-transitive graphs. In particular, we construct cores using kneser graphs K(n, 2n + 1). In the third part, we investigate the properties of a graph in terms of the directed graphs generated by the circular (k, d)-colorings and characterize the properties of vertex chic-critical and edge chi c-critical graphs, I also provide some sufficient conditions for chic(G) = chi( G). Circular clique number is also defined in terms of graph homomorphisms. Relations between circular clique number and clique number, as well as circular chromatic number are provided. |
