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Spectral radius and rainbow Hamiltonicity in bipartite graphs

Meng chen, Ruifang Liu, Qixuan Yuan

Abstract

Let $\mathcal{G}=\{G_1, G_2, \ldots , G_k\}$ be a family of bipartite graphs on the same vertex set. A rainbow Hamilton path (cycle) in $\mathcal{G}$ is a path (cycle) that visits each vertex precisely once such that any two edges belong to different graphs of $\mathcal{G}.$ In this paper, by adopting the technique of bi-shifting, we present tight sufficient conditions in terms of the spectral radius for a family $\mathcal{G}$ to admit a rainbow Hamilton path and cycle, respectively. Meanwhile, we completely characterize the corresponding spectral extremal graphs.

Spectral radius and rainbow Hamiltonicity in bipartite graphs

Abstract

Let be a family of bipartite graphs on the same vertex set. A rainbow Hamilton path (cycle) in is a path (cycle) that visits each vertex precisely once such that any two edges belong to different graphs of In this paper, by adopting the technique of bi-shifting, we present tight sufficient conditions in terms of the spectral radius for a family to admit a rainbow Hamilton path and cycle, respectively. Meanwhile, we completely characterize the corresponding spectral extremal graphs.
Paper Structure (5 sections, 27 theorems, 37 equations)

This paper contains 5 sections, 27 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm1.4']}
  4. Proof of theorem \ref{['thm1.5']}
  5. Proof of theorem \ref{['thm1.6']}

Key Result

Theorem 1.1

Let $G$ be a balanced bipartite graph on $2n$ vertices with minimum degree $\delta(G) \geq k,$ where $k \geq 0$ and $n \geq (k+2)^2.$ (i) If $k\neq 1$ and $\rho(G)\geq \rho(Q^k_n),$ then $G$ admits a Hamilton path unless $G\cong Q^k_n.$ (ii) If $k = 1$ and $\rho(G)\geq \rho(R^k_n),$ then $G$ admits

Theorems & Definitions (48)

  • Theorem 1.1: Li and Ningli2017spectral
  • Corollary 1.1: Li and Ningli2017spectral
  • Theorem 1.2: Li and Ningli2017spectral
  • Corollary 1.2: Li and Ningli2017spectral
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: Li and Ningli2016spectral
  • Corollary 1.3: Li and Ningli2016spectral
  • Theorem 1.6
  • Lemma 2.1: Csikváricsikvari2009conjecture
  • ...and 38 more