Bijections Between Smirnov Words and Hamiltonian Cycles in Complete Multipartite Graphs
El-Mehdi Mehiri
TL;DR
This work addresses the problem of counting undirected Hamiltonian cycles in complete multipartite graphs $K_{n,n,\ldots,n}$ by establishing a bijection with Smirnov words of length $mn$ having balanced letter multiplicities. The authors develop a color-word framework, derive an exact inclusion–exclusion formula for color-word counts, and translate these into exact counts of Hamiltonian cycles via $H_m(n)=\frac{W_m(n)}{2mn}=\frac{(n!)^m S_m(n)}{2mn}$. They further obtain asymptotic expressions using Stirling’s approximation, revealing factorial growth modulated by an adjacency-avoidance factor, and extend the theory to nonuniform part sizes $K_{n_1,\dots,n_m}$, directed graphs, and circular Smirnov words, thereby addressing a Knuth problem on nonuniform tripartite graphs. The results unify combinatorial word problems with graph enumeration, yielding closed formulas, asymptotics, and broad generalizations with potential applications in enumerative combinatorics and related algorithmic contexts.
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.