Okay, I think I can show that $\chi \leq 5$ for the second graph on verret's list.

Using Iglin's wonderful matlab package for graph theory (http://www.mathworks.com/matlabcentral/fileexchange/4266) I found out that its independence number is $\alpha=10$.

In a recent paper (http://www.sciencedirect.com/science/article/pii/S0012365X09002842) Kohl & Schiermeyer have shown that Reed's conjecture holds for graphs with $\Delta \geq n - \alpha -4$.

QED

P.S. There is a good chance that actually the 5 can be reduced to 4 as Reed's conecture is true without the ceiling (there's a good discussion in: Does every triangle-free graph with maximum degree at most 6 have a 5-colouring?). However, I haven't seen the Kohl & Schiermeyer proof yet and it might conceivably not carry over to the case without a ceiling.

Okay, I think I can show that $\chi \leq 5$ for the second graph on verret's list.

Using Iglin's wonderful matlab package for graph theory (http://www.mathworks.com/matlabcentral/fileexchange/4266) I found out that its independence number is $\alpha=10$.

In a recent paper (http://www.sciencedirect.com/science/article/pii/S0012365X09002842) Kohl & Schiermeyer have shown that Reed's conjecture holds for graphs with $\Delta \geq n - \alpha -4$.

QED

Okay, I think I can show that $\chi \leq 5$ for the second graph on verret's list.

Using Iglin's wonderful matlab package for graph theory (http://www.mathworks.com/matlabcentral/fileexchange/4266) I found out that its independence number is $\alpha=10$.

In a recent paper (http://www.sciencedirect.com/science/article/pii/S0012365X09002842) Kohl & Schiermeyer have shown that Reed's conjecture holds for graphs with $\Delta \geq n - \alpha -4$.

QED

Okay, I think I can show that $\chi \leq 5$ for the second graph on verret's list.

Using Iglin's wonderful matlab package for graph theory (http://www.mathworks.com/matlabcentral/fileexchange/4266) I found out that its independence number is $\alpha=10$.

In a recent paper (http://www.sciencedirect.com/science/article/pii/S0012365X09002842) Kohl & Schiermeyer have shown that Reed's conjecture holds for graphs with $\Delta \geq n - \alpha -4$.

QED

P.S. There is a good chance that actually the 5 can be reduced to 4 as Reed's conecture is true without the ceiling (there's a good discussion in: Does every triangle-free graph with maximum degree at most 6 have a 5-colouring?). However, I haven't seen the Kohl & Schiermeyer proof yet and it might conceivably not carry over to the case without a ceiling.