Choice numbers of graphs

@misc{citeulike:2395504, abstract = {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 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.}, author = {Gutner, Shai }, citeulike-article-id = {2395504}, eprint = {0802.2157}, keywords = {choice, graph}, month = {Feb}, posted-at = {2008-02-18 18:53:16}, priority = {2}, title = {Choice numbers of graphs}, url = {http://arxiv.org/abs/0802.2157}, year = {2008} }

TY - ELEC ID - citeulike:2395504 L3 - citeulike-article-id:2395504 TI - Choice numbers of graphs N2 - 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 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. KW - choice KW - graph AU - Gutner, Shai PY - 2008/Feb/15/ UR - http://arxiv.org/abs/0802.2157 ER -