| Summary: | For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><msub><mi>κ</mi><mi>G</mi></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> denotes the maximum number <i>k</i> of edge disjoint trees <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><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>k</mi></msub></mrow></semantics></math></inline-formula> in <i>G</i>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mrow><mo>(</mo><msub><mi>T</mi><mi>i</mi></msub><mo>)</mo></mrow><mo>∩</mo><mi>V</mi><mrow><mo>(</mo><msub><mi>T</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>=</mo><mi>S</mi></mrow></semantics></math></inline-formula> for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>≠</mo><mi>j</mi></mrow></semantics></math></inline-formula>. For an integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></mrow></semantics></math></inline-formula>, the generalized <i>r</i>-connectivity of <i>G</i> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>κ</mi><mi>r</mi></msub><mrow><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mo movablelimits="true" form="prefix">min</mo><mo>{</mo></mrow><msub><mi>κ</mi><mi>G</mi></msub><mrow><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mo>|</mo><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>and</mi><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mo>|</mo><mi>S</mi><mo>|</mo><mo>=</mo><mi>r</mi><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. In fact, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>κ</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the traditional connectivity of <i>G</i>. Hence, the generalized <i>r</i>-connectivity is an extension of traditional connectivity. The exchanged folded hypercube <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>F</mi><mi>H</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> are positive integers, is a variant of the hypercube. In this paper, we find that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>κ</mi><mn>3</mn></msub><mrow><mo>(</mo><mi>E</mi><mi>F</mi><mi>H</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mi>s</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mi>t</mi></mrow></semantics></math></inline-formula>.
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