Planar Turán number of the 7-cycle

2023/06/23 by Shi, Ruilin, Walsh, Zach, Yu, Xingxing · 3 citations
#05C10 #05C35 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2306.13594

Abstract

The planar Turán number \textrmex_{\mathcal P}(n,H) of a graph H is the maximum number of edges in an n-vertex planar graph without H as a subgraph. Let C_ℓ denote the cycle of length ℓ. The planar Turán number \textrmex_{\mathcal P}(n,C_ℓ) behaves differently for ℓ≤ 10 and for ℓ≥ 11, and it is known when ℓ ∈ \3,4,5,6\. We prove that \textrmex_{\mathcal P}(n,C₇) ≤ (18n)/(7) - (48)/(7) for all n > 38, and show that equality holds for infinitely many integers n.

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