Majority dynamics on random graphs: the multiple states case

We study the evolution of majority dynamics with more than two states on the binomial random graph $G(n,p)$. In this process, each vertex has a state in $\{1,\dots ,k\}$, with $k\ge 3$, and at each round every vertex adopts state $i$ if it has more neighbours in state $i$ that in any other state. Ties are resolved randomly. We show that with high probability the process reaches unanimity in at most three rounds, if $np\gg n^{2\mathrm{/}3}$.