From Colourful to Rainbow Paths in Graphs: Colouring the Vertices

Colourful connection concepts in graph theory such as rainbow connection, proper connection, odd connection or conflict-free connection have received a lot of attention. For an integer $$k \ge 1$$ we call a path P in a graph G k-colourful, if at least k vertices of P are pairwise differently coloured. A graph G is k-colourful connected, if any two vertices of G are connected by a k-colouful path. Now we call the least integer k, which makes G k-colourful connected, the k-colourful connection number of G. In this paper, we introduce the (strong, internal) k-colourful connection number of a graph, establish bounds for our new invariants in several graph classes as well as compute exact values for $$k\in [3]$$ .