Approaching Number Theory conjectures through Graph Theory
There are some results proven by graph theory, however I am not sure whether those results are good examples you want. Because number theory has offsprings by cross-breeding with some other fields, like arithmetic combinatorics and additive number theory 

In number theory, Szemerédi's theorem that was formerly the Erdős–Turán conjecture is a good example proven by graph theory. (Although it is not necessary to prove by graph theory. Other proofs are using different approaches, e.g., Hillel Furstenberg uses ergodic theory, and Timothy Gowers uses both Fourier analysis and combinatorics)

And this result leads to Green–Tao theorem.