Is there a graph $G$ such that $\chi(G)$ is finite, but there is no [total coloring](http://en.wikipedia.org/wiki/Total_coloring) with finitely many colours?



Graph with finite chromatic number but infinite total chromatic number
Yes there is: consider the complete bipartite graph $K\_{\omega,\omega}$, which has chromatic number 2, but every total coloring requires infinitely many colors.


