Sparse $4$-critical graphs have low circular chromatic number

TL;DR

This paper investigates the structure and edge bounds of 4-critical graphs with constraints on circular coloring, revealing new lower bounds and characterizing their Gallai Trees, especially in relation to odd cycles and wheel graphs.

Contribution

It establishes new edge bounds for 4-critical graphs with no (7,2)-coloring and characterizes their Gallai Tree structures, extending understanding of critical graphs with coloring restrictions.

Findings

Graphs with no (7,2)-coloring have at least 17v(G)/10 edges.

If Gallai Tree contains an odd cycle, the graph is an odd wheel.

Critical graphs with a large clique in Gallai Tree are complete graphs.

Abstract

Kostochka and Yancey proved that every $4$-critical graph $G$ has $e(G) \geq \frac{5v(G) - 2}{3}$, and that equality holds if and only if $G$ is $4$-Ore. We show that a question of Postle and Smith-Roberge implies that every $4$-critical graph with no $(7,2)$-circular-colouring has $e(G) \geq \frac{27v(G) -20}{15}$. We prove that every $4$-critical graph with no $(7,2)$-colouring has $e(G) \geq \frac{17v(G)}{10}$ unless $G$ is isomorphic to $K_{4}$ or the wheel on six vertices. We also show that if the Gallai Tree of a $4$-critical graph with no $(7,2)$-colouring has every component isomorphic to either an odd cycle, a claw, or a path. In the case that the Gallai Tree contains an odd cycle component, then $G$ is isomorphic to an odd wheel. In general, we show a $k$-critical graph with no $(2k-1,2)$-colouring that contains a clique of size $k-1$ in it's Gallai Tree is isomorphic to $K_{k}$.