Vertex coverings by coloured induced graphs — Frames and Umbrellas

A graph G homogeneously embeds in a graph H if for every vertex x of G and every vertex yof H there is an induced copy of G in H with x at y. The graph G uniformly embeds in H if for every vertex y of H there is an induced copy of G in H containing y. For positive integer k, let f_k(G) (respectively, g_k(G)) be the minimum order of a graph H whose edges can be k-coloured such that for each colour, G homogeneously embeds (respectively, uniformly embeds) in the graph given by V (H) and the edges of that colour. We investigate the values f₂(G) and g₂(G) for special classes of G, in particular when G is a star or balanced complete bipartite graph. Then we investigate f_k(G) and g_k(G) when k ≤ 3 and G is a complete graph.