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

Let $$G=(V(G),E(G))$$ G = ( V ( G ) , E ( G ) ) be a graph and s ,  t integers with $$s\le t$$ s ≤ t . If we can assign an s -subset $$\phi (v)$$ ϕ ( v ) of the set $$\{1, 2,\ldots ,t\}$$ { 1 , 2 , … , t } to each vertex v of V ( G ) such that $$\phi (u)\cap \phi (v)=\emptyset $$ ϕ ( u ) ∩ ϕ ( v ) = ∅ for every edge $$uv\in E(G)$$ u v ∈ E ( G ) , then G is called ( t  :  s )-colorable, and such an assignment $$\phi $$ ϕ is called a ( t  :  s )-coloring of G . Let $$C_n$$ C n denote a cycle of length n . In this paper, we show that every planar graph without $$C_4$$ C 4 and $$C_6$$ C 6 is (7 : 2)-colorable and thus has fractional chromatic number at most 7/2.