<p>The generalized connectivity is an extension of the traditional connectivity. It can measure the ability of a network <i>G</i> to connect any <i>s</i> vertices in <i>G</i>. For <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(S\subseteq V(G)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>⊆</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\kappa _G(S)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>κ</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denotes the maximum number <i>r</i> of internally disjoint trees <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(T_1,T_2,\ldots ,T_r\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>T</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>T</mi> <mi>r</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> in <i>G</i> such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(V(T_i)\cap V(T_j)=S\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mi>j</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(i,j\in \{1,2,\ldots ,r\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>r</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(i\ne j\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> </math></EquationSource> </InlineEquation>. For an integer <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(2\le k\le |V(G)|\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mo stretchy="false">|</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>, the generalized <i>k</i>-connectivity of <i>G</i>, denoted by <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\kappa _k(G)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>κ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, is defined as the minimum value of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\kappa _G(S)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>κ</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> over all <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(S\subseteq V(G)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>⊆</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(|S|=k\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>S</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation>, i.e., <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(\kappa _k(G)=\min \{\kappa _G(S)|S\subseteq V(G)\,\,\,\text {and}\,\,\, |S|=k\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>κ</mi> <mi>k</mi> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo movablelimits="true">min</mo> <mo stretchy="false">{</mo> </mrow> <msub> <mi>κ</mi> <mi>G</mi> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mi>S</mi> <mo>⊆</mo> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mtext>and</mtext> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mo stretchy="false">|</mo> <mi>S</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mi>k</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Exchanged crossed cube <i>ECQ</i>(<i>s</i>,&#xa0;<i>t</i>) is an important variation network of hypercube. In this paper, we get that <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(\kappa _4(ECQ(s,t))=s\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>κ</mi> <mn>4</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mi>C</mi> <mi>Q</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>s</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(3\le s\le t\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>≤</mo> <mi>s</mi> <mo>≤</mo> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

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The generalized 4-connectivity of exchanged crossed cube

  • Wantao Ning,
  • Rongshuan Geng

摘要

The generalized connectivity is an extension of the traditional connectivity. It can measure the ability of a network G to connect any s vertices in G. For \(S\subseteq V(G)\) S V ( G ) , \(\kappa _G(S)\) κ G ( S ) denotes the maximum number r of internally disjoint trees \(T_1,T_2,\ldots ,T_r\) T 1 , T 2 , , T r in G such that \(V(T_i)\cap V(T_j)=S\) V ( T i ) V ( T j ) = S for \(i,j\in \{1,2,\ldots ,r\}\) i , j { 1 , 2 , , r } and \(i\ne j\) i j . For an integer \(2\le k\le |V(G)|\) 2 k | V ( G ) | , the generalized k-connectivity of G, denoted by \(\kappa _k(G)\) κ k ( G ) , is defined as the minimum value of \(\kappa _G(S)\) κ G ( S ) over all \(S\subseteq V(G)\) S V ( G ) with \(|S|=k\) | S | = k , i.e., \(\kappa _k(G)=\min \{\kappa _G(S)|S\subseteq V(G)\,\,\,\text {and}\,\,\, |S|=k\}\) κ k ( G ) = min { κ G ( S ) | S V ( G ) and | S | = k } . Exchanged crossed cube ECQ(st) is an important variation network of hypercube. In this paper, we get that \(\kappa _4(ECQ(s,t))=s\) κ 4 ( E C Q ( s , t ) ) = s with \(3\le s\le t\) 3 s t .