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Choosability and fractional chromatic numbers. (English) Zbl 0877.05020
Let $$G=(V,E)$$ be a graph. The problem of finding its chromatic number, $$\chi(G)$$, can be formulated as an integer linear program: $\text{minimize}\quad \sum_{S\in{\mathcal S}(G)}\phi(S),\quad\text{over all }\phi\in{\mathcal P}(G);$ $\text{subject to}\quad\sum_{\phi\in S\in{\mathcal S}(G)}\phi(S)\geq 1,\quad\text{for all }\nu\in V,$ where $${\mathcal S}(G)$$ denotes the collection of all independent subsets of $$V$$ and $${\mathcal P}(G)$$ denotes the collection of all functions $$\phi:{\mathcal S}(G)\to \{0,1\}$$. The minimum is $$\chi(G)$$. If we replace $${\mathcal P}(G)$$ by $${\mathcal R}(G)$$, the collection of all functions $$\phi:{\mathcal S}(G)\to[0,1]$$, the minimum is called the fractional chromatic number $$\chi^*(G)$$.
Another variation of the chromatic number is the choice ratio of $$G$$ and it is defined as follows. For integers $$a$$ and $$b$$ with $$a\geq 2b>1$$, we say that $$G$$ is $$(a,b)$$-choosable if, for any assignment of lists of colors, $$L(\nu)$$, to the vertices of $$G$$ with $$|L(\nu)|=a$$ (for all $$\nu\in V$$), there are sublists $$C(\nu)\subset L(\nu)$$ with $$|C(\nu)|= b$$ (for all $$\nu\in V$$) and $$C(\nu)\cap C(\omega)=\varnothing$$, whenever $$\nu$$ and $$\omega$$ are adjacent. The choice ratio of $$G$$ is then defined to be $$\inf\{{a\over b}\mid G$$ is $$(a,b)$$-choosable}.
The authors give several results relating these two extensions of the chromatic number; the easiest to state is Theorem 1.2: The choice ratio of any graph equals its fractional chromatic number.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C35 Extremal problems in graph theory 05C38 Paths and cycles
##### Keywords:
choosability; fractional chromatic number; choice ratio
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##### References:
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