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.
A smallest trivalent graph with no 4, 8, or 10 cycles. A smallest trivalent graph with no 4, 8, or 12 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 450 vertices. I'd be interested to hear of any smaller examples (ge at cs dot indstate dot ee dee ewe).