Extremal $H$-Free Planar Graphs

Yongxin Lan,Zi-Xia Song,Yongtang Shi

doi:10.37236/8255

Yongxin Lan, Zi-Xia Song + Show 1 more

Open Access

https://doi.org/10.37236/8255

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Abstract

Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory 83 (2016) 213–230], that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define $ex_{_\mathcal{P}}(n,H)$ to be the maximum number of edges in an $H$-free planar graph on $n $ vertices. We first obtain several sufficient conditions on $H$ which yield $ex_{_\mathcal{P}}(n,H)=3n-6$ for all $n\ge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated Erdős-Stone Theorem. We then completely determine $ex_{_\mathcal{P}}(n,H)$ when $H$ is a wheel or a star. Finally, we examine the case when $H$ is a $(t, r)$-fan, that is, $H$ is isomorphic to $K_1+tK_{r-1}$, where $t\ge2$ and $r\ge 3$ are integers. However, determining $ex_{_\mathcal{P}}(n,H)$, when $H$ is a planar subcubic graph, remains wide open.

Highlights

All graphs considered in this paper are finite and simple
We show that every (K1 + 2K2)-free planar graph on n 5 vertices has at most 19n/8 − 4 edges
We show that every (K1 + 3K2)-free planar graph G on n 13 vertices has at most 17n/6 − 4 edges

Summary

Introduction

All graphs considered in this paper are finite and simple. We use Kt, Ct and Pt to denote the complete graph, cycle, and path on t vertices, respectively. By Proposition 3(f ), exP (n, H) remains unknown for K4-free planar graphs H with exactly one vertex, say u, of degree ∆(H) 6 and ∆(H[N (u)]) 2. It seems non-trivial to determine exP (n, H) for all such H. For every integer n 5, let On denote the unique outerplane graph with 2n − 3 edges, maximum degree 4, and the outer face of order n; let On be a different drawing of On with one unique inner face of order n; and let On∗ be the planar triangulation obtained from On and On by identifying the outer face of On with the unique n-face of On in such a way that On∗ is a simple graph. There does not exist a 4-regular planar graph on 7 vertices, or a 5-regular planar graph on 14 vertices, or a planar graph on n ∈ {11, 13} vertices with exactly one vertex of degree 4 and n − 1 vertices of degree 5

Proof of Proposition 3

Proof of Theorem 4

Proof of Theorem 5

Proof of Theorem 6

Concluding remarks