The Minimum Number of $4$-Cycles in a Maximal Planar Graph with Small Number of Vertices

TL;DR

This paper confirms a conjecture regarding the minimum number of 4-cycles in maximal planar graphs with small numbers of vertices, extending known bounds to all small cases.

Contribution

It proves the conjecture on the minimum number of 4-cycles in maximal planar graphs for all small vertex counts, completing previous partial results.

Findings

Confirmed the conjecture for small vertex counts

Established sharp lower bounds for all small cases

Extended previous results to complete the characterization

Abstract

Hakimi and Schmeichel determined a sharp lower bound for the number of cycles of length 4 in a maximal planar graph with $n$ vertices, $n\geq 5$. It has been shown that the bound is sharp for $n = 5,12$ and $n\geq 14$ vertices. However, the authors only conjectured the minimum number of cycles of length 4 for maximal planar graphs with the remaining small vertex numbers. In this note, we confirm their conjecture.