Abstract
We prove that every planar triangle-free graph on $n$ vertices has fractional chromatic number at most $3-3/(3n+1)$.
Highlights
The interest in the chromatic properties of triangle-free planar graphs originated with Grotzsch’s theorem [6], stating that such graphs are 3-colorable
Bollobas and Tucker [1] had conjectured that there is always a larger independent set, which was confirmed by Steinberg and Tovey [12] even in a stronger sense: all triangle-free planar n-vertex graphs admit a 3-coloring where not all color classes have the same size, and at least one of them forms an independent set of size at least n+1 3
We show that planar triangle-free graphs the electronic journal of combinatorics 22(4) (2015), #P4.11 of maximum degree 4 and without separating 4-cycles cannot have fractional chromatic number arbitrarily close to 3
Summary
The interest in the chromatic properties of triangle-free planar graphs originated with Grotzsch’s theorem [6], stating that such graphs are 3-colorable. Every planar triangle-free n-vertex graph of maximum degree at most four has fractional chromatic number at the graphs of Jones’s construction contain a large number of separating 4cycles (all their faces have length five). We show that planar triangle-free graphs the electronic journal of combinatorics 22(4) (2015), #P4.11 of maximum degree 4 and without separating 4-cycles cannot have fractional chromatic number arbitrarily close to 3. Dvorak and Mnich [5] proved that there exists β > 0 such that all planar triangle-free n-vertex graphs without separating 4-cycles contain an independent set of size at least n/(3 − β) This gives evidence that the restriction on the maximum degree in Theorem 3 might not be necessary. There exists δ > 0 such that every planar graph of girth at least five has fractional chromatic number at most 3 − δ
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