Improved bounds on coloring of graphs

Given a graph G with maximum degree @D>=3, we prove that the acyclic edge chromatic number a^'(G) of G is such that a^'(G)@[email protected]?9.62(@D-1)@?. Moreover we prove that: a^'(G)@[email protected]?6.42(@D-1)@? if G has girth g>=5; a^'(G)@[email protected]?5.77(@D-1)@? if G has girth g>=7; a^'(G)@[email protected]?4.52(@D-1)@? if g>=53; a^'(G)@[email protected]+2 if g>[email protected][email protected]@D(1+4.1/[email protected])@?. We further prove that the acyclic (vertex) chromatic number a(G) of G is such that a(G)@[email protected][email protected]^4^/^[email protected]@?. We also prove that the star-chromatic number @g"s(G) of G is such that @g"s(G)@[email protected][email protected]^3^/^[email protected]@?. We finally prove that the @b-frugal chromatic number @g^@b(G) of G is such that @g^@b(G)@[email protected]?max{k"1(@b)@D,k"2(@b)@D^1^+^1^/^@b/(@b!)^1^/^@b}@?, where k"1(@b) and k"2(@b) are decreasing functions of @b such that k"1(@b)@?[4,6] and k"2(@b)@?[2,5]. To obtain these results we use an improved version of the Lovasz Local Lemma due to Bissacot et al. (2011) [6].

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