Given a wireless ad hoc network and an end-to-end traffic pattern, the power rate function refers to the minimum total power required to support different throughput under a layered model of wireless networks. A critical notion of the layered model is the realizable graphs, which describe possible bit-rate supplies on the links by the physical and medium access layers. Under the layered model, the problem of finding the power rate function can be transformed into finding the minimum-power realizable graph that can provide a given throughput. We introduce a usage conflict graph to represent the conflicts among different uses of the wireless medium. Testing the realizability of a given graph can be transformed into finding the (vertex) chromatic number, i.e., the minimum number of colors required in a proper vertex-coloring, of the associated usage conflict graph. Based on an upper bound of the chromatic number, we propose a linear program that outputs an upper bound of the power rate function. A lower bound of the chromatic number is the clique number. We propose a systematic way of identifying cliques based on a geometric analysis of the space sharing among active links. This leads to another linear program, which yields a lower bound of the power rate function. We further apply greedy vertex-coloring to fine tune the bounds. Simulations results demonstrate that the obtained bounds are tight in the low power and low rate regime.