Fractional chromatic number, maximum degree and girth
Nombre chromatique fractionnaire, degré maximum et maille
Résumé
We prove new lower bounds on the independence ratio of graphs of maximum degree ∆ ∈ {3, 4, 5} and girth g ∈ {6,. .. , 12}, establishing notably that i(4, 10) ≥ 1/3 and i(5, 8) ≥ 2/7. We also demonstrate that every graph G of girth at least 7 and maximum degree ∆ has fractional chromatic number at most min (2∆+2 k−3 +k)/k over k∈N. In particular, the fractional chromatic number of a graph of girth 7 and maximum degree ∆ is at most (2∆+9)/5 when ∆ ∈ [3, 8], at most (∆+7)/3 when ∆ ∈ [8, 20], at most (2∆+23)/7 when ∆ ∈ [20, 48], and at most ∆/4 + 5 when ∆ ∈ [48, 112].
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