Let G be a graph. The spectral radius of G is the largest eigenvalue of its adjacency matrix. For a non-complete bipartite graph G with parts X and Y, the bipartite toughness of G is defined as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$t^{B}(G)=\min \left\{ \frac{|S|}{c(G-S)}\right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>t</mi> <mi>B</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo movablelimits="true">min</mo> <mfenced close="}" open="{"> <mfrac> <mrow> <mo stretchy="false">|</mo> <mi>S</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo>-</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, where the minimum is taken over all proper subsets <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$S\subset X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>⊂</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> (or <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$S\subset Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>⊂</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation>) such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$c(G-S)>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo>-</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we give a sharp spectral radius condition for balanced bipartite graphs G with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$t^{B}(G)\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>t</mi> <mi>B</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> to guarantee that G contains Hamilton cycles. This solves a problem proposed in [<CitationRef CitationID="CR9">9</CitationRef>].

A spectral condition for Hamilton cycles in tough bipartite graphs

期刊论文
发表日期：

2025年10月15日

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Lianyang Ai,
Wenqian Zhang

摘要

Let G be a graph. The spectral radius of G is the largest eigenvalue of its adjacency matrix. For a non-complete bipartite graph G with parts X and Y, the bipartite toughness of G is defined as $t^{B}(G)=\min \left\{ \frac{|S|}{c(G-S)}\right\} $ $t^{B} (G) = min \{\frac{| S |}{c (G - S)}\}$ , where the minimum is taken over all proper subsets $S\subset X$ $S \subset X$ (or $S\subset Y$ $S \subset Y$ ) such that $c(G-S)>1$ $c (G - S) > 1$ . In this paper, we give a sharp spectral radius condition for balanced bipartite graphs G with $t^{B}(G)\ge 1$ $t^{B} (G) \geq 1$ to guarantee that G contains Hamilton cycles. This solves a problem proposed in [9].