Graphs Without Cycles of Specified Lengths

The following graphs are motivated by the Erdos/Gyarfas conjecture which asserts (more or less) that every graph with minimum degree at least 3 has a cycle whose length is a power of 2.

	A smallest trivalent graph with no 4, 6, or 10 cycles.		A trivalent graph of order 32 with no 4, 8 or 32 cycles.
	A smallest trivalent graph with no 4 or 8 cycles (there are 3 others)		Another example of a graph with no cycles of lengths 4 and 8.
	The (unique) smallest trivalent graph with no 4 or 6 cycles.		A smallest trivalent graph with no 4, 6 or 8 cycles.

The smallest graph I know of with no 4, 8 or 16 cycles has 78 vertices. The smallest with no 4, 8, 16 or 32 cycles has 540 vertices. I'd be interested to hear of any smaller examples.