Hamiltonian decomposition in graphs: when probability isn't zero or one?

Imagine generating thousands of random graphs from a structure called a step-graphon. Now ask yourself: how many of these graphs contain a Hamiltonian decomposition? A known result states that, for almost all cases, the probability of finding one converges to either zero or one. But what happens when it doesn’t? This is the so-called residual case. A team of researchers has shown that even in this situation, the probability still converges to a certain limit. And more: they found an explicit expression to calculate it. The proof is complete and the results were verified through numerical simulations. This kind of study is important to better understand complex networks, used in biology, computer science, and social media. The mathematics behind graphs helps model relationships between objects, people, or data. That’s why understanding when a Hamiltonian decomposition exists is crucial: it means finding a path that touches each node exactly once. A seemingly abstract result, but with concrete applications in network optimization and algorithms.

The research focuses on a specific property of so-called step-graphons, which are limits of sequences of graphs defined on measure spaces. These objects allow the study of the asymptotic behavior of large networks. A Hamiltonian decomposition requires that a graph can be partitioned into cycles covering all nodes. The main result shows that, in the residual case, the probability of observing such a decomposition converges to a well-defined value, expressible in terms of the geometric properties of the graphon. Numerical validation confirms the accuracy of the theoretical formula. This work opens new perspectives in the probabilistic analysis of networks and computational complexity theory.