Fibonacci Identities and Graph Colorings

We generalize both the Fibonacci and Lucas numbers to the context of graph colorings, and prove some identities involving these numbers. As a corollary we obtain new proofs of some known identities involving Fibonacci numbers such as $$F_{r+s+t}=F_{r+1}{F}_{s+1}{F}_{t+1}+F_r{F}_s{F}_t-F_{r-1}{F}_{s-1}{F}_{t-1}\mathrm{.}$$