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Stiebitz et al(2006), Yu(2008) and Kurt(2009) has separately shown $ \chi'(G) \geq \Delta+\sqrt{\Delta/2} $ implies Goldberg Conjecture. While Yu's method gives a methodological approach to the general problem, Kurt provides a very short and elementary proof.

Scheide (2008) has shown $ \chi'(G)>\frac{15}{14}\Delta+\frac{12}{14} $ implies the Goldberg Conjecture.

Kurt(2009) has shown $ \chi'(G)>\frac{17}{16}\Delta+\frac{14}{16} $ implies the Goldberg Conjecture.

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