Choosability of toroidal graphs without short cycles

Let G be a toroidal graph without cycles of a fixed length k , and χ l ( G ) the list chromatic number of G . We establish tight upper bounds of χ l ( G ) for the following values of k :\documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}\begin{eqnarray*}\chi_l({{G}})\le \left\{\begin{array}{ll}{{4}} & \mbox{if } {{k}}\in \{{{3}},{{4}},{{5}}\} \\{{5}} & \mbox{if } {{k}} = {{6}} \\{{6}} & \mbox{if } {{k}} = {{7}}\end{array}\right.\end{eqnarray*}\end{document} © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 1–15, 2010.