图的选择数
Choice numbers of graphs
摘要：
通过证明如果图$G$是$(a:b)$ -可选的，且$c/d > a/b$是可选的，则$G$不一定是$(c:d)$ -可选的，得到了Erdés，Rubin和Taylor问题的一个解.最简单的情况下，另一个问题，由同一作者，解决，证明每个2-choosable图也是$(4:2)$ -choosable。应用概率方法，给出了图的$k^{th}$选择数的一个上界。证明了具有最大出度$d$且无奇有向圈的有向图对每个$k \geq 1$都是$(k(d+1):k)$ -可选的.本文给出的其他结果与图的强选择数（强色数的推广）有关。我们的结论与复杂性分析的一些决策问题有关的图的可选性。
英文摘要：
A solution to a problem of Erdős, Rubin and Taylor is obtained by showing that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not necessarily $(c:d)$-choosable. The simplest case of another problem, stated by the same authors, is settled, proving that every 2-choosable graph is also $(4:2)$-choosable. Applying probabilistic methods, an upper bound for the $k^{th}$ choice number of a graph is given. We also prove that a directed graph with maximum outdegree $d$ and no odd directed cycle is $(k(d+1):k)$-choosable for every $k \geq 1$. Other results presented in this article are related to the strong choice number of graphs (a generalization of the strong chromatic number). We conclude with complexity analysis of some decision problems related to graph choosability.
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