Exact generalized Turán number forversus suspension of

Sayan Mukherjee

Abstract: Let P4<math><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msub></math> denote the path graph on 4 vertices. The suspension of P4<math><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msub></math>, denoted by Pˆ4<math><msub is="true"><mrow is="true"><mover accent="true" is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mo is="true">ˆ</mo></mrow></mover></mrow><mrow is="true"><mn is="true">4</mn></mrow></msub></math>, is the graph obtained via adding an extra vertex and joining it to all four vertices of P4<math><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msub></math>. In this note, we demonstrate that for n≥8<math><mi is="true">n</mi><mo is="true">≥</mo><mn is="true">8</mn></math>, the maximum number of triangles in any n-vertex graph not containing Pˆ4<math><msub is="true"><mrow is="true"><mover accent="true" is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mo is="true">ˆ</mo></mrow></mover></mrow><mrow is="true"><mn is="true">4</mn></mrow></msub></math> is ⌊n2/8⌋<math><mo stretchy="true" is="true">⌊</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo stretchy="false" is="true">/</mo><mn is="true">8</mn><mo stretchy="true" is="true">⌋</mo></math>. Our method uses simple induction along with computer programming to prove a base case of the induction hypothesis.