Planar graphs without 4- and 6-cycles are (7 : 2)-colorable

Abstract

Let $G=(V(G),E(G))$ be a graph and s, t integers with $s\le t$. If we can assign an s-subset $\phi (v)$ of the set $\{1, 2,\ldots ,t\}$ to each vertex v of V(G) such that $\phi (u)\cap \phi (v)=\emptyset$ for every edge $uv\in E(G)$, then G is called (t : s)-colorable, and such an assignment $\phi$ is called a (t : s)-coloring of G. Let $C_n$ denote a cycle of length n. In this paper, we show that every planar graph without $C_4$ and $C_6$ is (7 : 2)-colorable and thus has fractional chromatic number at most 7/2.