Sum choice numbers of some graphs

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Sum choice numbers of some graphs

Let f be a function assigning list sizes to the vertices of a graph G. The sum choice number of G is the minimum ∑ v∈V (G) f(v) such that for every assignment of lists to the vertices of G, with list sizes given by f , there exists proper coloring of G from the lists. We answer a few questions raised in a paper of Berliner, Bostelmann, Brualdi, and Deaett. Namely, we determine the sum choice nu...

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A solution to a problem of Erd\H{o}s, 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 nu...

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ژورنال

عنوان ژورنال: Discrete Mathematics

سال: 2009

ISSN: 0012-365X

DOI: 10.1016/j.disc.2008.04.044