Coloring graphs with dense neighborhoods

It is shown that any graph with maximum degree $\Delta$ in which the average degree of the induced subgraph on the set of all neighbors of any vertex exceeds $\frac{6k^2}{6k^2 + 1}\Delta + k + 6$ is either $(\Delta - k)$-colorable or contains a clique on more than $\Delta - 2k$ vertices. In the $k=1$ case we improve the bound on the average degree to $\frac23\Delta + 4$ and the bound on the clique number to $\Delta-1$. As corollaries, we show that every graph satisfies $\chi \leq \max\set{\omega, \Delta - 1, 4\alpha}$ and every graph satisfies $\chi \leq \max\set{\omega, \Delta - 1, \ceil{\frac{15 + \sqrt{48n + 73}}{4}}}$.