Rainbow hamiltonicity and the spectral radius

Let G(G1.1... Gn) be a family of graphs of order n with the same vertex set. A rainbow Hamiltonian cycle in G is a cycle that visits each vertex precisely once such that any two edges belong to different graphs of G. We show that if each G(1) has more than ("2") +1 edges, then admits a rainbow Hamiltonian cycle and pose the problem of characterizing rainbow Hamiltonicity under the condition that all G(1) have at least ((1)) +1 edges. Towards a solution of that problem, we give a sufficient condition for the existence of a rainbow Hamiltonian cycle in terms of the spectral radii of the graphs in G and completely characterize the corresponding extremal graphs.<br /> (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining. Al training, and similar technologies.</span>