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Planar graphs with the maximum number of induced 6-cycles

Michael Savery

Abstract

For large $n$ we determine the maximum number of induced 6-cycles which can be contained in a planar graph on $n$ vertices, and we classify the graphs which achieve this maximum. In particular we show that the maximum is achieved by the graph obtained by blowing up three pairwise non-adjacent vertices in a 6-cycle to sets of as even size as possible, and that every extremal example closely resembles this graph. This extends previous work by the author which solves the problem for 4-cycles and 5-cycles. The 5-cycle problem was also solved independently by Ghosh, Győri, Janzer, Paulos, Salia, and Zamora.

Planar graphs with the maximum number of induced 6-cycles

Abstract

For large we determine the maximum number of induced 6-cycles which can be contained in a planar graph on vertices, and we classify the graphs which achieve this maximum. In particular we show that the maximum is achieved by the graph obtained by blowing up three pairwise non-adjacent vertices in a 6-cycle to sets of as even size as possible, and that every extremal example closely resembles this graph. This extends previous work by the author which solves the problem for 4-cycles and 5-cycles. The 5-cycle problem was also solved independently by Ghosh, Győri, Janzer, Paulos, Salia, and Zamora.

Paper Structure

This paper contains 6 sections, 11 theorems, 10 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and outline of the paper
  3. Preliminaries
  4. Large planar graphs contain a vertex in few induced 6-cycles
  5. Proof of Theorem \ref{['thm:main']}
  6. Longer even cycles

Key Result

Theorem 3

For large $n$, $G$ is an $n$-vertex planar graph containing $f_I(n,C_6)$ induced 6-cycles if, and only if, $G\in \mathcal{F}_n$. Hence for large $n$,

Figures (4)

  • Figure 1: The graphs $F_{n,m}$ and $F'_{n,m}$ in different cases
  • Figure 2: A graph in $\mathcal{F}_n$
  • Figure 3: Some graphs used in the proof of Claim \ref{['claim:8_in_int']}
  • Figure 4: Two graphs used in the proof of Lemma \ref{['lem:n_equiv_1']}

Theorems & Definitions (34)

  • Conjecture 1: savery2021planar
  • Definition 2
  • Theorem 3
  • Lemma 4
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Corollary 8
  • ...and 24 more