Sub-exponentially many 3-colorings of triangle-free planar graphs

TL;DR

This paper improves the lower bound on the number of 3-colorings in triangle-free planar graphs from sub-exponential to a larger sub-exponential scale, specifically involving a square root of n.

Contribution

It establishes a significantly stronger lower bound on the number of 3-colorings for triangle-free planar graphs, advancing Thomassen's conjecture.

Findings

Number of 3-colorings is at least 2^sqrt(n/362)

Improves previous lower bound from 2^[n^(1/12)/20000]

Supports the conjecture of exponential growth in colorings

Abstract

Thomassen conjectured that every triangle-free planar graph on n vertices has exponentially many 3-colorings, and proved that it has at least 2^[n^(1/12)/20000] distinct 3-colorings. We show that it has at least 2^sqrt(n/362) distinct 3-colorings.