Planar graphs with the maximum number of induced 6-cycles

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.

23 pages, 4 figures; final version; v2: small changes incorporating referee's comments

References in corpus (5)

• Subgraph densities in a surface
• The Maximum Number of Pentagons in a Planar Graph
• Homomorphism counts in robustly sparse graphs
• Planar graphs with the maximum number of induced 4-cycles or 5-cycles
• The maximum number of odd cycles in a planar graph