SIAM Journal on Discrete Mathematics, Volume 34, Issue 1, Page 538-555, January 2020.
A graph $G$ is $(a:b)$-colorable if there exists an assignment of $b$-element subsets of $\{1,\ldots,a\}$ to vertices of $G$ such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph $G$ and a vertex $x\in V(G)$, the graph $G$ has a set coloring $\varphi$ by subsets of $\{1,\ldots,6\}$ such that $|\varphi(v)|\geq 2$ for $v\in V(G)$ and $|\varphi(x)|=3$. As a corollary, every triangle-free planar graph on $n$ vertices is $(6n:2n+1)$-colorable. We further use this result to prove that for every $\Delta$, there exists a constant $M_{\Delta}$ such that every planar graph $G$ of girth at least five and maximum degree $\Delta$ is $(6M_{\Delta}:2M_{\Delta}+1)$-colorable. Consequently, planar graphs of girth at least five with bounded maximum degree $\Delta$ have fractional chromatic number at most $3-\frac{3}{2M_{\Delta}+1}$.

中文翻译：

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围长五的平面图的分数着色

SIAM离散数学杂志，第34卷，第1期，第538-555页，2020年1月。
如果存在$ \ {1，\ ldots，a \} $的$ b $个元素子集分配给$ G $的顶点，则图$ G $是$（a：b）$可着色的到相邻顶点不相交。我们首先显示出，对于每个无三角形的平面图$ G $和V（G）$中的顶点$ x \，图$ G $都有一组着色子\\ varphi $，其中子集$ \ {1，\ ldots ，6 \} $，使得$ | \ varphi（v）| \ geq 2 $等于$ v \ in V（G）$和$ | \ varphi（x）| = 3 $。作为推论，$ n $个顶点上的每个无三角形平面图都是$（6n：2n + 1）$可着色的。我们进一步使用该结果证明，对于每个$ \ Delta $，存在一个常数$ M _ {\ Delta} $，这样每个周长至少为5且最大度数$ \ Delta $的平面图$ G $为$（6M_ {\ Delta}：2M _ {\ Delta} +1）$（可着色）。所以，