Planar graphs with the maximum number of induced 6-cycles

TL;DR

This paper determines the maximum number of induced 6-cycles in large planar graphs, classifies extremal graphs achieving this maximum, and extends previous work on smaller cycles.

Contribution

It identifies the extremal structure for the maximum induced 6-cycles in planar graphs, building on prior results for 4- and 5-cycles.

Findings

Maximum number of induced 6-cycles in large planar graphs identified

Extremal graphs are blow-ups of 6-cycle with specific vertex sets

The extremal structure closely resembles the constructed blow-up graph

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, Gy\H{o}ri, Janzer, Paulos, Salia, and Zamora.