Cover Time of the Complete Graph K₄

A simple random walk moves on the complete graph $K_4$. At each step, the walker moves to one of the $3$ neighbors uniformly at random.

(a) Compute the maximum hitting time $t_{\mathrm{hit}} = \max_{u,v} h(u \to v)$.

(b) Using Matthews' theorem, which gives $$t_{\mathrm{hit}} \cdot H_{n-1} \ge E[\tau_{\mathrm{cov}}] \ge t_{\mathrm{hit}} \cdot H_{n-1} \cdot \frac{n-1}{n},$$ where $n = 4$ and $H_k = \sum_{i=1}^k 1/i$, bound the expected cover time.

(c) Compute $E[\tau_{\mathrm{cov}}]$ exactly by decomposing into phases: after visiting $k$ distinct vertices, find the expected time to discover the $(k+1)$-th.