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 ≥ 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 ∈ [ 3 ] .