A Candidate Counterexample to L(N) = f(N) in the Divisor Graph: The Case N = 111

For the divisor graph induced by [N] = {1, …, N}, let F(N) denote the minimum number of paths in a vertex partition, let L(N) be the maximum length of a path appearing in an optimal partition, and let f(N) be the unrestricted longest-path length. Melotti and Saias asked whether L(N) = f(N) for every positive integer N. We present the case N = 111, for which explicit path witnesses and exact integer certificates yield F(111) = 21 and L(111) = 85 < 86 = f(111). Thus, N = 111 provides a candidate counterexample to the proposed equality. The argument combines an explicit optimal path partition, degree-based certificates for maximum-degree-two subgraphs, structural reductions obtained by deleting the vertices {1, 2, 3}, and exhaustive separator and integer-weight certificates for the remaining boundary configurations. As a methodological complement, we also establish L(N) = f(N) for 34 ≤ N ≤ 37, extending the previously known range through 37. The cases 38 ≤ N ≤ 110 are not addressed, so no claim is made that 111 is the least counterexample.