Density of 5/2-critical graphs

TL;DR

This paper establishes a lower bound on the density of 5/2-critical graphs, characterizes those achieving the bound, and applies the result to show certain planar graphs are 5/2-colorable.

Contribution

It proves a minimum edge count for 5/2-critical graphs and identifies all graphs that meet this bound, extending understanding of graph colorability.

Findings

Minimum edges for 5/2-critical graphs: (5n-2)/4

Complete characterization of extremal 5/2-critical graphs

Planar graphs with girth at least 10 are 5/2-colorable

Abstract

A graph G is 5/2-critical if G has no circular 5/2-coloring (or equivalently, homomorphism to C_5), but every proper subgraph of G has one. We prove that every 5/2-critical graph on n>=4 vertices has at least (5n-2)/4 edges, and list all 5/2-critical graphs achieving this bound. This implies that every planar or projective-planar graph of girth at least 10 is 5/2-colorable.