We prove the simple-undirected form of the Kannan–Tetali–Vempala conjecture: the switch chain on simple undirected graphs mixes in polynomial time for every graphical degree sequence. For a lazy chain that proposes switches uniformly on four vertices, the total-variation mixing time at distance 1/4 is at most $2n^8$. We also give an exactly uniform sampler for every graphical labeled degree vector. It uses unbiased random bits, terminates almost surely, and has expected polynomial bit running time.