2025/03/07 by Zhu, Xiutao, Wang, Xiaolin, Zhang, Yanbo +1 · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.05166
The Turán number \ex(n,H) is the maximum number of edges that an n-vertex H-free graph can have. The suspension \widehatH is obtained from H by adding a new vertex which is adjacent to all vertices of H and a tree is balanced if the sizes of its two color classes differ at most 1. In this paper, we obtain a sharp bound of \ex(n,\widehatT) when n≥ 4(4k)6 based on the Erdős-Sós conjecture. We also show the bound is sharp for infinitely many n and characterize all extremal graphs. In particular, if T satisfies some conditions such as T contains a matching covering all vertices in one color class, then the bound is sharp for all n. This is a new class of graphs whose decomposition family does not contain a linear forest but we still can determine its Turán number.