论文标题
图表的分数盖上的注释
A note on fractional covers of a graph
论文作者
论文摘要
图$ g $的分数着色是一个函数，该函数将非负实际值分配给所有可能的$ g $的$ g $，其中包含$ g $的任何顶点，因此每个顶点的这些值的总和至少为一个。分数色数是分数着色分配的值的最小总和，而所有可能的颜色为$ g $。由Bosica和Tardif引入的分数盖是分数颜色的扩展，因此，实价函数在属于给定的图形类别的所有可能的$ g $的所有可能子图上作用。事实证明，分数色数是分数盖号的特殊实例。在这项工作中，我们调查了作用于$（k+1）$ - $ g $的$（k+1）$的子图的分数封面，尽管这些子图与$ k $ g $的分数封面共享一些相似之处，但它们表现出一些特殊性。我们首先表明，如果简单的图$ g_2 $是简单图$ g_1 $的同构图像，则在$（k+1）$ - $ g_1 $的无clique-free子图上定义的分数封面编号在上面以相应的$ g_2 $为界。我们利用此结果来获取相关的分数封面范围的界限，这些图形是$ n $ - 颜色或$ a \！\！\！：\！\！b $ -olourable。
A fractional colouring of a graph $G$ is a function that assigns a non-negative real value to all possible colour-classes of $G$ containing any vertex of $G$, such that the sum of these values is at least one for each vertex. The fractional chromatic number is the minimum sum of the values assigned by a fractional colouring over all possible such colourings of $G$. Introduced by Bosica and Tardif, fractional covers are an extension of fractional colourings whereby the real-valued function acts on all possible subgraphs of $G$ belonging to a given class of graphs. The fractional chromatic number turns out to be a special instance of the fractional cover number. In this work we investigate fractional covers acting on $(k+1)$-clique-free subgraphs of $G$ which, although sharing some similarities with fractional covers acting on $k$-colourable subgraphs of $G$, they exhibit some peculiarities. We first show that if a simple graph $G_2$ is a homomorphic image of a simple graph $G_1$, then the fractional cover number defined on the $(k+1)$-clique-free subgraphs of $G_1$ is bounded above by the corresponding number of $G_2$. We make use of this result to obtain bounds for the associated fractional cover number of graphs that are either $n$-colourable or $a\!\!:\!\!b$-colourable.
