论文标题
重新审视Kempe等效列表着色
Kempe Equivalent List Colorings Revisited
论文作者
论文摘要
在颜色$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a和$ b $的组件中，由颜色$ a $ a $ a $ a $ a $ a $ a $ a $和$ b $的组成部分。 A \ Emph {Kempe Change}是互换某些Kempe链的颜色的操作。对于列表分配$ l $和$ l $ - 颜色$φ$，如果执行kempe更改会产生另一个$ l $ coloring，则Kempe更改为\ emph {$ L $ -VALID}，对于$φ$。如果我们可以通过$ L $ -L $ -VALID KEMPE更改的序列形成一个$ l $ -olorings \ emph {$ l $ equivalent}。 a \ emph {学位分配}是一个列表分配$ l $，因此$ l（v）\ ge d（v）$ in V（g）$中的每个$ v \ y。 Cranston和Mahmoud（\ emph {combinatorica}，2023）问：哪个图形$ g $和学位分配$ l $的$ g $的$ g $是$ g $的所有$ l $ g $ as $ l $ is $ l $均等吗？我们证明，对于每个4个连接的图$ g $，它都不完整，每个学位分配$ l $ $ g $，所有$ l $ g $的颜色均为$ g $，均为$ l $ - 等值。
A \emph{Kempe chain} on colors $a$ and $b$ is a component of the subgraph induced by colors $a$ and $b$. A \emph{Kempe change} is the operation of interchanging the colors of some Kempe chain. For a list-assignment $L$ and an $L$-coloring $φ$, a Kempe change is \emph{$L$-valid} for $φ$ if performing the Kempe change yields another $L$-coloring. Two $L$-colorings are \emph{$L$-equivalent} if we can form one from the other by a sequence of $L$-valid Kempe changes. A \emph{degree-assignment} is a list-assignment $L$ such that $L(v)\ge d(v)$ for every $v\in V(G)$. Cranston and Mahmoud (\emph{Combinatorica}, 2023) asked: For which graphs $G$ and degree-assignment $L$ of $G$ is it true that all the $L$-colorings of $G$ are $L$-equivalent? We prove that for every 4-connected graph $G$ which is not complete and every degree-assignment $L$ of $G$, all $L$-colorings of $G$ are $L$-equivalent.
