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- W3046202471 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 $K_{k}$." @default.
- W3046202471 created "2020-08-03" @default.
- W3046202471 creator A5008362514 @default.
- W3046202471 date "2020-07-30" @default.
- W3046202471 modified "2023-09-27" @default.
- W3046202471 title "Sparse $4$-critical graphs have low circular chromatic number" @default.
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