Abstrakt
A graph G is (a : b)-colorable if there exists an assignment of b-element subsets of {1, ..., a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph G and a vertex x is an element of V(G), the graph G has a set coloring phi by subsets of {1, ..., 6} such that vertical bar phi(v)vertical bar >= 2 for v is an element of V(G) and vertical bar phi(x)vertical bar = 3.
As a corollary, every triangle-free planar graph on n vertices is (6n : 2n + 1)-colorable. We further use this result to prove that for every Delta, there exists a constant M-Delta such that every planar graph G of girth at least five and maximum degree Delta is (6M(Delta) : 2M(Delta) + 1)-colorable.
Consequently, planar graphs of girth at least five with bounded maximum degree Delta have fractional chromatic number at most 3 -3/2M(Delta)+1.