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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A uniform set covering lemma


Author: David W. Matula
Journal: Proc. Amer. Math. Soc. 48 (1975), 255-261
MSC: Primary 05B40
MathSciNet review: 0376408
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Abstract: The bounded set system $ H = (V,\mathfrak{F})$ is composed of a nonvoid set $ V$ and a set, $ \mathfrak{F}$, of nonvoid subsets of $ V$, a finite number of which cover $ V$. $ C \subset V$ is a critical subset of $ H$ if every proper subset of $ C$ requires fewer members of $ \mathfrak{F}$ to cover it than are needed to cover $ C$. For $ \vert\mathfrak{F}\vert$ finite, it is shown that every $ A \subset V$ contains a critical $ C \subset A$ requiring the same number of members of $ \mathfrak{F}$ in a minimum cover. For $ v \in V,l(v)$ is the largest number of members of $ \mathfrak{F}$ in any minimum cover of any critical set containing $ v$. For $ \vert\mathfrak{F}\vert$ finite, it is shown that there exists a covering $ {A_1},{A_2}, \cdots ,{A_k},{A_i} \in \mathfrak{F}$ for $ 1 \leq i \leq k$, such that $ v \in \bigcup\nolimits_{i = 1}^{l(v)} {{A_i}} $ for all $ v \in V$. An application to graph coloring is described.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0376408-5
PII: S 0002-9939(1975)0376408-5
Keywords: Set coverings, critical sets, minimum covers, uniform cover, hypergraph, graph covering, Grundy function
Article copyright: © Copyright 1975 American Mathematical Society