2025/09/04 by Hei, Doudou, Hou, Xinmin, Ma, Yue
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.03851
Given graphs H and F, the generalized Turán number \ex(n, H, F) is defined as the maximum number of copies of H in an n-vertex graph that contains no copy of F. The suspension \widehatF of a graph F is obtained by adding a new vertex that is adjacent to every vertex of F. Mubayi and Mukherjee (2023, DM) conjectured that \ex(n, K3, \widehatPk)=\lfloor (k-2)/(2)\rfloor ⋅ (n2)/(8)+o(n2), where Pk is a path on k≥ 4 vertices. Using the triangle removal lemma, they verified this conjecture for k=4,5,6. Later, Mukherjee (2024, DM) established the exact value \ex(n, K3, \widehatP4)=\lfloor n2/8\rfloor. In this paper, using the stability method, we determine the exact value of \ex(n, K3, \widehatP5) by showing that for sufficiently large n, \ex(n,K3, \widehatP5)=\lfloor n2/8\rfloor.