🤖 AI Summary
This paper addresses the exact enumeration of Hamiltonian cycles in complete multipartite graphs. Methodologically, it constructs an explicit bijection between Hamiltonian cycles and Smirnov words—sequences over a finite alphabet with no adjacent identical letters—under a letter-frequency balance condition; this bijection is then generalized to non-uniform multipartite graphs, and combined with the inclusion–exclusion principle to derive a closed-form formula for the number of Hamiltonian cycles. Using Stirling’s approximation and asymptotic analysis, the work further obtains a precise logarithmic asymptotic expansion, revealing factorial-order growth. The contributions unify combinatorial enumeration of constrained words and graph-theoretic cycle counting, establishing a cross-domain analytical framework that bridges adjacency-constrained structures and Hamiltonicity problems.
📝 Abstract
We establish a bijective correspondence between Smirnov words with balanced letter multiplicities and Hamiltonian paths in complete $m$-partite graphs $K_{n,n,ldots,n}$. This bijection allows us to derive closed inclusion-exclusion formulas for the number of Hamiltonian cycles in such graphs. We further extend the enumeration to the generalized nonuniform case $K_{n_1,n_2,ldots,n_m}$. We also provide an asymptotic analysis based on Stirling's approximation, which yields compact factorial expressions and logarithmic expansions describing the growth of the number of Hamiltonian cycles in the considered graphs. Our approach unifies the combinatorial study of adjacency-constrained words and the enumeration of Hamiltonian cycles within a single analytical framework.