[Paper Review] Spectral radius and Hamiltonicity of graphs with large minimum degree: revisited

Jun Ge, Bo Ning|arXiv (Cornell University)|Jun 28, 2016

Graph theory and applications18 references3 citations

TL;DR

This paper establishes spectral radius conditions for Hamiltonicity in graphs and balanced bipartite graphs with large minimum degree. It proves that for sufficiently large $ n $, a graph $ G $ of order $ n $ with $ \delta(G) \geq k $ and spectral radius $ \lambda(G) \geq n-k-1 $ is Hamiltonian unless it is isomorphic to one of two specific exceptional graphs, extending Nikiforov's result and providing a spectral analogue of Moon-Moser's theorem.

ABSTRACT

Investigating the relationship between eigenvalues of a graph and its cycle structure is a standard topic in spectral graph theory. This paper mainly concerns spectral conditions for Hamilton cycles in graphs and in balanced bipartite graphs, respectively. Our main results are written as: (1) Let $k\geq 1$, $n\geq \max\{\frac{1}{2}k^3+k+4,6k+5\}$, and let $G$ be a graph of order $n$, with minimum degree $\delta(G)\geq k$. If its spectral radius $\lambda(G)\geq n-k-1$, then $G$ has a Hamilton cycle unless $G=K_1\vee (K_k+K_{n-k-1})$, or $G=K_k\vee (kK_1+K_{n-k-1})$. (2) Let $k\geq 1$, $n\geq k^3+2k+4$, and let $G$ be a balanced bipartite graph of order $2n$, with minimum degree $\delta(G)\geq k$. If its spectral radius $\lambda(G)\geq \sqrt{n(n-k)}$, then $G$ has a Hamilton cycle unless $G=B^k_n$, where $B^k_n$ is the graph obtained from $K_{n,n}$ by deleting a $K_{k,n-k}$. Our first theorem shows that a very recent theorem of Nikiforov still holds when $n<k^3$ (if $k$ is sufficiently large). Our second result further gives a spectral analogue of Moon-Moser's theorem on Hamilton cycles in balanced bipartite graphs, and extends a previous result due to Li and one of the authors here for $n$ sufficiently large.

Motivation & Objective

• To extend Nikiforov's recent spectral condition for Hamiltonicity to smaller values of $ n $ when $ k $ is large.
• To provide a spectral analogue of Moon-Moser's theorem on Hamilton cycles in balanced bipartite graphs.
• To characterize the exceptional graphs that satisfy the spectral radius condition but are not Hamiltonian.
• To generalize previous results by Li and one of the authors for large $ n $.

Proposed method

• Use spectral graph theory to analyze the relationship between the largest eigenvalue (spectral radius) and the existence of Hamilton cycles.
• Apply extremal graph constructions, particularly the join of complete graphs, to identify potential counterexamples.
• Establish tight bounds on $ n $ in terms of $ k $ to ensure the spectral radius condition implies Hamiltonicity.
• Analyze the structure of the exceptional graphs $ K_1\vee (K_k+K_{n-k-1}) $ and $ K_k\vee (kK_1+K_{n-k-1}) $, and their bipartite analogues.
• Use the spectral radius condition $ \lambda(G) \geq \sqrt{n(n-k)} $ in the balanced bipartite case to derive Hamiltonicity.
• Compare the results with known theorems such as Nikiforov's and Moon-Moser's to establish the novelty and strength of the spectral conditions.

Experimental results

Research questions

• RQ1 Under what spectral radius condition does a graph with minimum degree $ \delta(G) \geq k $ guarantee a Hamilton cycle, even when $ n < k^3 $?
• RQ2 What are the precise exceptional graphs that satisfy the spectral radius condition but are not Hamiltonian?
• RQ3 Can a spectral condition be established for Hamiltonicity in balanced bipartite graphs that mirrors Moon-Moser's theorem?
• RQ4 How does the spectral radius threshold $ \lambda(G) \geq n-k-1 $ relate to the existence of Hamilton cycles in dense graphs with high minimum degree?
• RQ5 To what extent do the results extend prior work by Li and the authors for large $ n $?

Key findings

• For $ n \geq \max\{\frac{1}{2}k^3+k+4, 6k+5\} $, a graph $ G $ of order $ n $ with $ \delta(G) \geq k $ and $ \lambda(G) \geq n-k-1 $ is Hamiltonian unless $ G $ is isomorphic to $ K_1\vee (K_k+K_{n-k-1}) $ or $ K_k\vee (kK_1+K_{n-k-1}) $.
• The spectral radius condition $ \lambda(G) \geq n-k-1 $ ensures Hamiltonicity even when $ n < k^3 $, extending Nikiforov's result to a broader range of $ n $.
• For balanced bipartite graphs of order $ 2n $ with $ \delta(G) \geq k $, the condition $ \lambda(G) \geq \sqrt{n(n-k)} $ guarantees a Hamilton cycle unless $ G = B^k_n $, the graph obtained by deleting a $ K_{k,n-k} $ from $ K_{n,n} $.
• The exceptional graph $ B^k_n $ is the only non-Hamiltonian graph satisfying the spectral radius condition in the balanced bipartite setting.
• The results provide a spectral analogue of Moon-Moser's theorem on Hamilton cycles in balanced bipartite graphs.
• The paper confirms that the spectral radius threshold $ \lambda(G) \geq \sqrt{n(n-k)} $ is sharp for Hamiltonicity in balanced bipartite graphs with minimum degree $ k $.