Volume 35 • Number 4 • 2012   • Equitable Coloring and Equitable Choosability of Planar Graphs Without 5- and 7-Cycles Aijun Dong, Xin Zhang and Guojun Li Abstract. A graph $G$ is equitably $k$-choosable if for any $k$-uniform list assignment $L$, $G$ is $L$-colorable and each color appears on at most $\lceil|V(G)|/k\rceil$ vertices. A graph $G$ is equitable $k$-colorable if $G$ has a proper vertex coloring with $k$ colors such that the size of the color classes differ by at most 1. In this paper, we prove that if $G$ is a planar graph without $5$- and $7$-cycles, then $G$ is equitably $k$-choosable and equitably $k$-colorable where $k\geq\max\{\Delta(G),7\}$. 2010 Mathematics Subject Classification: 05C15. Full text: PDF