Summary: A binary Gray code \(G(n)\) of length \(n\), is a list of all \(2^n\) \(n\)-bit codewords such that successive codewords differ in only one bit position. The sequence of bit positions where the single change occurs when going to the next codeword in \(G(n)\), denoted by \(S(n):= s_1,s_2,\dots, s_{2^n-1}\), is called the transition sequence of the Gray code \(G(n)\). The graph \({\mathcal G}_{G(n)}\) induced by a Gray code \(G(n)\) has vertex set \(\{1,2,\dots,n\}\) and edge set \(\{\{s_i,s_{i+1}\}: 1\leq i\leq 2^n-2\}\). If the first and the last codeword differ only in position \(s_{2^n}\), the code is cyclic and we extend the graph by two more edges \(\{s_{2^n-1},s_{2^n}\}\) and \(\{s_{2^n},s_1\}\). We solve a problem of E. L. Wilmer and M. D. Ernst [Discrete Math. 257, 585–598 (2002; Zbl 1009.05094)] about a construction of an \(n\)-bit Gray code inducing the complete graph \(K_n\). The technique used to solve this problem is based on a Gray code construction due to Bakos [A. Ádám, Truth functions and the problem of their realization by two-terminal graphs, Budapest: Akadémial Kiadó (1968; Zbl 0155.02003)], and which is presented in D. E. Knuth [The art of computer programming. Vol. 4, Fascicle 2. Generating all tuples and permutations. Reading, MA: Addison-Wesley (2005; Zbl 1127.68068)].