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<p>We study Hamiltonian circles in the doubly signed complete graph <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Sigma _n = (K_n, \sigma , {\mathbb {F}}_2^2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Σ</mi> <mi>n</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <msubsup> <mi mathvariant="double-struck">F</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. A circle’s double sign is defined as the sum of its edge labels. I establish conditions under which Hamiltonian circles realize all four possible double signs and prove that this occurs when the set of triangle double signs contains at least three distinct values. The proof is based on an analysis of triangle bases of the binary cycle space, structural properties of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(K_4\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation> subgraphs, and explicit Hamiltonian constructions.</p>

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Double signs of Hamiltonian circles in doubly signed complete graphs

  • 期刊论文

  • 发表日期：

    2026年7月1日

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  • Xiyong Yan

摘要

We study Hamiltonian circles in the doubly signed complete graph \(\Sigma _n = (K_n, \sigma , {\mathbb {F}}_2^2)\) Σ n = ( K n , σ , F 2 2 ) . A circle’s double sign is defined as the sum of its edge labels. I establish conditions under which Hamiltonian circles realize all four possible double signs and prove that this occurs when the set of triangle double signs contains at least three distinct values. The proof is based on an analysis of triangle bases of the binary cycle space, structural properties of \(K_4\) K 4 subgraphs, and explicit Hamiltonian constructions.