Fractional Coloring of Planar Graphs of Girth Five

Abstract

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 ∈ V (G), the graph G has a set coloring ϕ by subsets of {1, . . . , 6} such that |ϕ(v)| ≥ 2 for v ∈ V (G) and |ϕ(x)| = 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 ∆, there exists a constant M∆ such that every planar graph G of girth at least five and maximum degree ∆ is (6M∆ : 2M∆ + 1)-colorable. Consequently, planar graphs of girth at least five with bounded maximum degree ∆ have fractional chromatic number at most 3 − 3 2M ∆ +1 .

Fractional vertex-arboricity of planar graphs

“…On the other hand, considerable attention has been given to generalizing bounds on the independence number to the fractional chromatic number. We can see these two themes, even just for planar graphs, in [5,7,8,9,10,12,17].…”