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arXiv · 2510.26597

Bijections Between Smirnov Words and Hamiltonian Cycles in Complete Multipartite Graphs

El-Mehdi Mehiri
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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.

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El-Mehdi Mehiri. 2025-10-30. Bijections Between Smirnov Words and Hamiltonian Cycles in Complete Multipartite Graphs. https://arxiv.org/abs/2510.26597

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