Improved bounds on coloring of graphs

Given a graph G with maximum degree Δ ≥ 3 , we prove that the acyclic edge chromatic number a ′ ( G ) of G is such that a ′ ( G ) ≤ ⌈ 9.62 ( Δ − 1 ) ⌉ . Moreover we prove that: a ′ ( G ) ≤ ⌈ 6.42 ( Δ − 1 ) ⌉ if G has girth g ≥ 5 ; a ′ ( G ) ≤ ⌈ 5.77 ( Δ − 1 ) ⌉ if G has girth g ≥ 7 ; a ′ ( G ) ≤ ⌈ 4.52 ( Δ − 1 ) ⌉ if g ≥ 53 ; a ′ ( G ) ≤ Δ + 2 if g ≥ ⌈ 25.84 Δ log Δ ( 1 + 4.1 / log Δ ) ⌉ . We further prove that the acyclic (vertex) chromatic number a ( G ) of G is such that a ( G ) ≤ ⌈ 6.59 Δ 4 / 3 + 3.3 Δ ⌉ . We also prove that the star-chromatic number χ s ( G ) of G is such that χ s ( G ) ≤ ⌈ 4.34 Δ 3 / 2 + 1.5 Δ ⌉ . We finally prove that the β -frugal chromatic number χ β ( G ) of G is such that χ β ( G ) ≤ ⌈ max { k 1 ( β ) Δ , k 2 ( β ) Δ 1 + 1 / β / ( β ! ) 1 / β } ⌉ , where k 1 ( β ) and k 2 ( β ) are decreasing functions of β such that k 1 ( β ) ∈ [ 4 , 6 ] and k 2 ( β ) ∈ [ 2 , 5 ] . To obtain these results we use an improved version of the Lovász Local Lemma due to Bissacot et al. (2011) [6].