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<p>The genus of a graph is the minimum genus (number of "handles") of an orientable surface in which the graph can be embedded without any edges crossing. A fundamental optimization problem in network topology research is to determine genus of a graph, which is a quantitative measure of a graph’s deviation from planarity. In this study, we determine the genus of the Cartesian products <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(K_{12t+7} \Box C_{2s}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mrow> <mn>12</mn> <mi>t</mi> <mo>+</mo> <mn>7</mn> </mrow> </msub> <mo>□</mo> <msub> <mi>C</mi> <mrow> <mn>2</mn> <mi>s</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(K_{12t+7} \Box P_{s}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mrow> <mn>12</mn> <mi>t</mi> <mo>+</mo> <mn>7</mn> </mrow> </msub> <mo>□</mo> <msub> <mi>P</mi> <mi>s</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(t \in \mathbb {Z}^+ \cup \{0\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> <mo>∪</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(s \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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On the Genus of Cartesian products of complete graph \(K_{12t+7}\) with cycles and paths

  • 期刊论文

  • 发表日期：

    2026年4月13日

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  • Jyoti Anant Pulgam,
  • Prashant Malavadkar

摘要

The genus of a graph is the minimum genus (number of "handles") of an orientable surface in which the graph can be embedded without any edges crossing. A fundamental optimization problem in network topology research is to determine genus of a graph, which is a quantitative measure of a graph’s deviation from planarity. In this study, we determine the genus of the Cartesian products \(K_{12t+7} \Box C_{2s}\) K 12 t + 7 □ C 2 s and \(K_{12t+7} \Box P_{s}\) K 12 t + 7 □ P s , where \(t \in \mathbb {Z}^+ \cup \{0\}\) t ∈ Z + ∪ { 0 } and \(s \ge 2\) s ≥ 2 .