论文标题
$ g $ free图的极端结果
Extremal results on $G$-free colorings of graphs
论文作者
论文摘要
令$ h =（v（h），e（h））$为图。 $ k $ - 颜色的$ h $是映射$π：v（h）\ longrightArrow \ {1,2，\ ldots，k \} $，以便每个颜色类都诱导$ k_2 $ free-free子图。对于图$ g $的订单，至少$ 2 $，$ g $ -free $ k $ - 颜色为$ h $的映射$π：v（h）\ longrightArrow \ {1,2，\ ldots，k \} $，以便由$ h $的$ h $的子类别由每种颜色的$ h $类别$ f $ g $ g $ g $ f $ fefe. $ g $ free $ h $的$ h $是最低数字$ k $，因此有$ g $ - free $ k $ - 颜色为$ h $，用$χ_g（h）$表示。图形$ h $是唯一的$ k $ - $ g $ - 如果$χ_g（h）= k $和每个$ k $ - $ g $ -free $ h $的$ h $的颜色都会产生相同的颜色类别。对于$ g $ - free或$ g $ -free-Minimal，图形$ h $是最小的，如果对于$ e（h）$的每个边缘，我们都有$χ_g（h \ setMinus \ {e \}）=χ_g（h）-1 $。在本文中，我们提供了一些界限和属性，涉及$ k $ - $ g $ - 免费着色和$ k $ - $ g $ -free-rinimal。
Let $H=(V(H),E(H))$ be a graph. A $k$-coloring of $H$ is a mapping $π: V(H) \longrightarrow \{1,2,\ldots, k\}$ so that each color class induces a $K_2$-free subgraph. For a graph $G$ of order at least $2$, a $G$-free $k$-coloring of $H$ is a mapping $π: V(H) \longrightarrow \{1,2,\ldots,k\}$ so that the subgraph of $H$ induced by each color class of $π$ is $G$-free, i.e. contains no copy of $G$. The $G$-free chromatic number of $H$ is the minimum number $k$ so that there is a $G$-free $k$-coloring of $H$, denoted by $χ_G(H)$. A graph $H$ is uniquely $k$-$G$-free colouring if $χ_G(H)=k$ and every $k$-$G$-free colouring of $H$ produces the same color classes. A graph $H$ is minimal with respect to $G$-free, or $G$-free-minimal, if for every edges of $E(H)$ we have $χ_G(H\setminus\{e\})= χ_G(H)-1$. In this paper we give some bounds and attribute about uniquely $k$-$G$-free colouring and $k$-$G$-free-minimal.
