| Summary: | For a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></semantics></math></inline-formula> and a set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> of a size at least 2, a path in <i>G</i> is said to be an <i>S</i>-path if it connects all vertices of <i>S</i>. Two <i>S</i>-paths <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>2</mn></msub></semantics></math></inline-formula> are said to be <i>internally disjoint</i> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mrow><mo>(</mo><msub><mi>P</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>∩</mo><mi>E</mi><mrow><mo>(</mo><msub><mi>P</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>=</mo><mo>∅</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mrow><mo>(</mo><msub><mi>P</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>∩</mo><mi>V</mi><mrow><mo>(</mo><msub><mi>P</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>=</mo><mi>S</mi></mrow></semantics></math></inline-formula>; that is, they share no vertices and edges apart from <i>S</i>. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>π</mi><mi>G</mi></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> denote the maximum number of internally disjoint <i>S</i>-paths in <i>G</i>. The <i>k</i>-path-connectivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>π</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <i>G</i> is then defined as the minimum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>π</mi><mi>G</mi></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <i>S</i> ranges over all <i>k</i>-subsets of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>. In this paper, we study the <i>k</i>-path-connectivity of the complete balanced tripartite graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></semantics></math></inline-formula> and obtain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>π</mi><mi>k</mi></msub><mfenced separators="" open="(" close=")"><msub><mi>K</mi><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></mfenced><mo>=</mo><mfenced separators="" open="⌊" close="⌋"><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mfenced></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>.</mo></mrow></semantics></math></inline-formula>
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