Computational Verification of the Buratti--Horak--Rosa Conjecture for Small Integers and Inductive Approaches

This paper presents a comprehensive computational approach to verify and inductively construct Hamiltonian paths for the Buratti--Horak--Rosa (BHR) Conjecture. The conjecture posits that for any multiset $L$ of $p-1$ positive integers not exceeding $\lfloor p/2 \rfloor$, there exists a Hamiltonian path in the complete graph $K_p$ with vertex-set $\{0, 1, \dots, p-1\}$ whose edge lengths (under the cyclic metric) match $L$, if and only if for every divisor $d$ of $p$, the number of multiples of $d$ appearing in $L$ is at most $p - d$. Building upon prior computational work by Mariusz Meszka, which verified the conjecture for all primes up to $p=23$, our Python program extends this verification significantly. We approach the problem by systematically generating frequency partitions (FPs) of edge lengths and employing a recursive backtracking algorithm. We report successful computational verification for all frequency partitions for integers $p < 32$, specifically presenting results for $p=31$ and a composite $p=26$. For the composite number $p=30$, the Python code took approximately 11 hours to verify on a Lenovo laptop. For $p=16$, $167,898$ valid multisets were processed, taking around 20 hours on Google Colab Pro+. Furthermore, we introduce and implement two constructive, inductive strategies for building Hamiltonian paths: (1) increasing the multiplicity of an existing edge length, and (2) adding a new edge length. These methods, supported by a reuse-insertion heuristic and backtracking search, demonstrate successful constructions for evolving FPs up to $p=40$. Through these empirical tests and performance metrics, we provide strong computational evidence for the validity of the BHR conjecture within the scope tested, and outline the scalability of our approach for higher integer values.