论文标题
在广义列表上$ \ g $ - 图形的无色颜色
On generalized list $\G$-free colorings of graphs
论文作者
论文摘要
对于给定的图形$ h $和图形属性$ p $，条件色的数字$χ（h，p）$ $ h $是最小的$ k $，因此可以将$ v（h）$分解为集合$ v_1，v_1，v_2，v_2，\ ldots，\ ldots，v_k $，v_k $，其中$ h [v_i] $满足$ p $ $ $ $ $ $ $ $ 1 \ pe \ pe fe pe fe pe fe pe fe pe fe pe \ pe fe pe \ pe \ pe fe pe fe pe fe pe。当属性$ p $为每个颜色类别不包含$ g $的副本时，我们编写$χ_{g}（h）$而不是$χ（g，p）$，这称为$ g $ - free-free chalomation number。因此，我们说$ h $具有$ k $ - $ g $ - 如果有地图$ c：v（h）\ longrightArrow \ {1，\ ldots，k \} $，以便每种$ c $ be $ g $ g $ - free。假设每个顶点$ v $ a Graph $ h $都分配了一组$ l（v）$的颜色，称为颜色列表。集合$ g（l）= \ {g（v）：v \ in V（h）\} $，那是$ g $以下$ h $的顶点选择的一组颜色。 $ l $ - 颜色$ g $称为$ g $ - free，因此： \ begin {inatizize} \ item $ g（v）\在l（v）$中，对于v（h）$中的任何$ v \。 \ item $ h [v_i] $是$ g $ - $ i = 1,2，\ ldots，l $。 \ end {inatizize}如果存在$ l $ - 颜色为$ h $，则$ h $称为$ l $ - $ g $ - free-colorable。图$ h $据说为$ k $ - $ g $ -free-choosable，如果存在$ l $ - 颜色的任何列表分配$ l $满足$ | l（v）| \ geq k $ in v（h）$中的$ v \ geq k $，$ h [v_i] $ h [v_i] $ be $ g $ be $ g $ be $ g $ for $ g $ for $ g $ for $ g $ for $ i = 1,2,2,2,2,2,2,2,2,2,2,2，2，2，2，让图$ h $和一系列图形$ \ g $，$ h $的$χ_ {\ g}^l（h）$是最后一个整数$ k $，因此$ h $是$ h $ as $ k $ - $ \ g $ -free-free-choosable-free-choosable-choosable-y. $ h [v_i] $ $ h [v_i] $ \ g $ k $ k $ k $ k $ $ \ g $的任何成员的副本。在本文中，我们表明某些图$ h $和$ g $，$χ_g^l（h）=χ_g（h）$，$χ_g^l（h \ oplus h'）\ leq uqχ_g^l（h）+χ_g^l（h）+χ_g^l（h''（h''）$ g $ g $，$ h $，以及$ h $和$ h $。另外，我们表明$χ_ {\ g}（h \ oplus k_n）=χ^l _ {\ g}（h \ oplus k_n）$，其中$ \ g $是所有$ d $ regratiargular grange grablings of-g $ and $ n $。
For given graph $H$ and graphical property $P$, the conditional chromatic number $χ(H,P)$ of $H$, is the smallest number $k$, so that $V(H)$ can be decomposed into sets $V_1,V_2,\ldots, V_k$, in which $H[V_i]$ satisfies the property $P$, for each $1\leq i\leq k$. When property $P$ be that each color class contains no copy of $G$, we write $χ_{G}(H)$ instead of $χ(G,P)$, which is called the $G$-free chromatic number. Due to this, we say $H$ has a $k$-$G$-free coloring if there is a map $c : V(H) \longrightarrow \{1,\ldots,k\}$, so that each of the color classes of $c$ be $G$-free. Assume that for each vertex $v$ of a graph $H$ is assigned a set $L(V)$ of colors, called a color list. Set $g(L) = \{g(v): v\in V(H)\}$, that is the set of colors chosen for the vertices of $H$ under $g$. An $L$-coloring $g$ is called $G$-free, so that: \begin{itemize} \item $g(v)\in L(v)$, for any $v\in V(H)$. \item $ H[V_i]$ is $G$-free for each $i=1,2,\ldots, L$. \end{itemize} If there exists an $L$-coloring of $H$, then $H$ is called $L$-$G$-free-colorable. A graph $H$ is said to be $k$-$G$-free-choosable if there exists an $L$-coloring for any list-assignment $L$ satisfying $|L(V)|\geq k$ for each $v\in V(H)$, and $H[V_i]$ be $G$-free for each $i=1,2,\ldots, L$. Let graph $H$ and a collection of graphs $\G$ are given, the $χ_{\G}^L(H)$ of $H$ is the last integer $k$, so that $H$ is $k$-$\G$-free-choosable i.e. $H[V_i]$ is $\G$-free for each $i=1,2,\ldots, k$ i.e. contains no copy of any member of $\G$. In this article, we show that $χ_G^L(H)=χ_G(H)$ for some graph $H$ and $G$, $χ_G^L(H\oplus H')\leq χ_G^L(H)+χ_G^L(H')$ for each $G$, $H$, and $H'$. Also, we show that $χ_{\G}(H\oplus K_n)=χ^L_{\G}(H\oplus K_n)$, where $\G$ is a collection of all $d$-regular graphs, and some $n$.
