2020/04/29 by Debarun Ghosh, Ervin Győri, Ghosh, Debarun +7 · 4 citations
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2004.14094
openalex publication_date 2020/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \rm exP(n,T,H) denote the maximum number of copies of T in an n-vertex planar graph which does not contain H as a subgraph. When T=K2, \rm exP(n,T,H) is the well studied function, the planar Turán number of H, denoted by \rm exP(n,H). The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both \rm exP(n,C4) and \rm exP(n,C5). Later on, Y. Lan, et al. continued this topic and proved that \rm exP(n,C6)≤ (18(n-2))/(7). In this paper, we give a sharp upper bound \rm exP(n,C6) ≤ (5)/(2)n-7, for all n≥ 18, which improves Lan's result. We also pose a conjecture on \rm exP(n,Ck), for k≥ 7.