Rainbow Hamiltonicity and the spectral radius

TL;DR

The paper investigates rainbow Hamiltonian cycles in a family \\mathcal{G}=\\{G_1,\\dots, G_n\\} on a common vertex set, extending classical Hamiltonicity results to rainbow transversals. It develops two main sufficient conditions: (i) an Ore-type size bound where each \\!e(G_i) > \\binom{n-1}{2}+1\\ guarantees a rainbow Hamiltonian cycle, and (ii) a spectral-radius bound with \\rho(G_i) > n-2\\ yielding the same conclusion except in the extremal isomorphism class \\cong K_1\\vee (K_{n-2}\\cup K_1).\\ The Kelmans transformation is employed to reduce to threshold-graph extremals, enabling precise extremal analyses, including a parallel result for the signless Laplacian radius with \\rho_S(G_i) > 2n-4\\. These results extend classical Hamiltonicity criteria to rainbow settings and illuminate the structure of extremal graph families under spectral constraints.

Abstract

Let $\mathcal{G}=\{G_1,\ldots,G_n \}$ be a family of graphs of order $n$ with the same vertex set. A rainbow Hamiltonian cycle in $\mathcal{G}$ is a cycle that visits each vertex precisely once such that any two edges belong to different graphs of $\mathcal{G}$. We show that if each $G_i$ has more than $\binom{n-1}{2}+1$ edges, then $\mathcal{G}$ admits a rainbow Hamiltonian cycle and pose the problem of characterizing rainbow Hamiltonicity under the condition that all $G_i$ have at least $\binom{n-1}{2}+1$ 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 $\mathcal{G}$ and completely characterize the corresponding extremal graphs.

Figures (3)

• Figure 1: Graph family with $G_1 = G_2=\cdots= G_n \cong K_1\vee (K_{n-2}\cup K_1)$ that is not rainbow Hamiltonian.
• Figure 2: Switches in Lemma \ref{['shift_ham']}
• Figure 3: Rainbow Hamiltonian cycles in Proposition \ref{['extgraphlem']}

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof