For integers \(r \ge 2\) and \(g \ge 3\) , an (r, g)-graph is an r-regular graph with girth g, and an (r, g)-cage is an (r, g)-graph of minimum order. It is conjectured that all (r, g)-cages with even g are bipartite, that is, have chromatic number 2. Here we introduce the idea of an \((r,g,\chi )\) -graph, an r-regular graph with girth g and chromatic number \(\chi \) . We investigate the existence of such graphs and study in detail the (r, 3, 3)-graphs of minimum order. We also consider \((r,g,\chi )\) -graphs for which there is a \(\chi \) -coloring where the color classes differ by at most 1.
