On Subgraphs Induced by Transversals in Vertex-Partitions of Graphs

For a fixed graph $H$ on $k$ vertices, we investigate the graphs, $G$, such that for any partition of the vertices of $G$ into $k$ color classes, there is a transversal of that partition inducing $H$. For every integer $k\geq 1$, we find a family ${\cal F}$ of at most six graphs on $k$ vertices such that the following holds. If $H\notin {\cal F}$, then for any graph $G$ on at least $4k-1$ vertices, there is a $k$-coloring of vertices of $G$ avoiding totally multicolored induced subgraphs isomorphic to $H$. Thus, we provide a vertex-induced anti-Ramsey result, extending the induced-vertex-Ramsey theorems by Deuber, Rödl et al.