Recent hardware demonstrations and advances in circuit compilation have made quantum computing with higher-dimensional systems (qudits) on near-term devices an attractive possibility. Some problems have more natural or optimal encodings using qudits over qubits. We explore this potential by formulating graph 3-coloring, a well-known and difficult problem with practical applications, using qutrits, and solve it using the quantum approximate optimization algorithm (QAOA). Qutrit-based cost and mixer Hamiltonians are constructed along with appropriate quantum circuits using qutrit gates. We run noiseless simulations using PennyLane to compare the formulation against qubit-based QAOA, and analyze the solution quality and resources required. Preliminary results show that the qutrit encoding finds more accurate solutions with a comparable set of hyperparameters, uses half as many qudits, and has a notably smaller circuit depth per layer than an efficient qubit encoding. This work suggests that qutrits may be useful in solving some problems on near-term devices, however further work is required to assess their potential in a noisy environment.