I tried to find a solution to this in the web but couldn't. Can you tell me if the following sentence is correct or else give me a counterexample?
$G$ is $4$-colorable if and only if each sub-graph $G'$ in $G$ is not isomorphic to $K_5$. At first glance it seems to be related to the four color theorem but it is not exactly a planar graph (e.g. $3\times 3$ complete bipartite graph) so it is not identical to FCT. Any ideas?