Can carbon and titanium form an ionic bond?
Yes, carbon can form ionic bonds with Titanium. $\ce{TiC}$ is titanium carbide, an ionic compound with $\ce{C^{4-}}$ (the carbide anion) and $\ce{Ti^4+}$ (titanium cation). It has the same crystal structure as $\ce{NaCl}$ (which probably has to do with the $\ce{1\!\!:\!1}$ ratio of anion to cation).

This is a space filling model of titanium carbide:



Here is sodium chloride for a crystal comparison (click the image for a larger version):



The second model (the $\ce{NaCl}$) shows the ionic bonds as covalent bonds but still, both titanium carbide and sodium chloride have a cubic lattice structure.

However, carbon does not often form ionic bonds with metals since it prefers to be covalent (much like oxygen). In fact carbon has much more covalent character than oxygen does which is why oxides form easily (just $\ce{O2 + element}$ and usually, oxides will form) but carbides don't (a catalyst is needed to form carbides and some oxides like gold oxide).