论文标题
稀疏$ 4 $ - 临界图的圆形色数低
Sparse $4$-critical graphs have low circular chromatic number
论文作者
论文摘要
Kostochka和Yancey证明，每$ 4 $ clatch $ g $都具有$ e（g）\ geq \ frac {5v（g）-2} {3} $，并且当$ g $是$ 4 $ -ORE时，该公平只能保持。我们表明，Postle和Smith-Roberge的问题意味着，每$ 4 $ - 临界图，没有$（7,2）$ - 圆形色的$ E（g）\ geq \ geq \ frac \ frac {27v（g）-20} -20} {15} $。我们证明，除非$ g $ no $ e（g）\ geq \ frac {17v（g）} {10} $，除非$ g $是同构为$ k_ {4} $，否则每$ 4 $ - 临界图，没有$（7,2）$ - 颜色具有$ e（g）\ geq \ frac {17v（g）} {10} $。我们还表明，如果没有$（7,2）$的$ 4 $批判图的加莱树具有每个组件同构的奇数循环，爪子或路径。如果加莱树包含一个奇数的循环组件，则$ g $对奇数车轮是同构的。通常，我们显示一个$ k $ - 批判图，没有$（2K-1,2）$ - 颜色包含一个大小$ k-1 $的大小为$ k-1 $的颜色，是同构为$ k_ {k {k {k} $。
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 $K_{k}$.
