Proof of Locke’s Conjecture, I

Locke’s conjecture (L) states that the binary hypercube Qn with k deleted vertices of each parity is Hamiltonian if n ≥ k + 2. In 2003, S. C. Locke and R. Stong published in The American Mathematical Monthly a proof of (L) for the case k = 1. In 2007, in the paper Path coverings with prescribed ends in faulty hypercubes the authors proved (L) for every k ≤ 4 and formulated the following conjecture (CG): Let n ≥ k + 3 and F be a set of k even (odd) and k+1 odd (even) vertices of Qn. If u, v are two even (odd) vertices of Qn − F then there exists a Hamiltonian path of Qn−F that connects u and v. (CG) is known to be true for every k ≤ 3 and every n ≥ k + 3. In this paper we prove that if n ≥ 7, 5 ≤ k ≤ n − 2 and (L) is true for every n1 ≤ n − 1 and every k1 ≤ n1 − 2 and (CG) is true for every n1 ≤ n − 1 and every k1 ≤ n1 − 3 then (L) is also true for n and k. To keep the paper shorter the proof that if n ≥ 7, 4 ≤ k ≤ n − 3 and (L) is true for every n1 ≤ n − 1 and every k1 ≤ n1− 2 and (CG) is true for every n1 ≤ n− 1 and every k1 ≤ n1 − 3 then (CG) is also true for n and k will appear in a forthcoming paper. In that way the two papers together complete the proofs of (L) and (CG).