Rainbow Vertex Coloring

In the Rainbow Vertex Coloring (RVC) problem we want to decide whether the vertices of a given graph can be colored with at most k colors so that the graph becomes rainbow vertex-connected. This problem is known to be NP-complete even in very restricted scenarios, and very few efficient algorithms are known for it. Rainbow coloring of a wheel graph, with three colors.

Every two non-adjacent vertices can be connected by a rainbow path, either directly through the center vertex (bottom left) or by detouring around one triangle to avoid a repeated edge color (bottom right). In the Rainbow Vertex Coloring (RVC) problem we want to decide whether the vertices of a given graph can be colored with at mostkcolors so that the graph becomes rainbow vertex-connected. This problem is known to be NP-complete even in very restricted scenarios, and very few efficient algorithms are known for it.

A proper vertex coloring of a connected graph G that results in a vertex rainbow-connected graph is a vertex rainbow coloring of G. The minimum number of colors needed in a vertex rainbow coloring of G is the vertex rainbow connection number vrc (G) of G. Thus if G is a connected graph of order n ≥ 2, then 2 ≤ vrc (G) ≤ n.

The Rainbow Vertex Coloring (RVC) problem takes as input a graph G and an integer k and asks whether G has a coloring with k colors under which it is rainbow vertex-connected. The rainbow vertex connection number of a graph G is the smallest number of colors needed in one such coloring and is denoted rvc(G). The minimum amount of colors assigned over all rainbow colorings that result from rainbow vertex antimagic labelings of G is the rainbow vertex antimagic connection number of G, rvac (G).

A vertex-colored graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists an x-y rainbow vertex-cut. In this case, the vertex-coloring c is called a rainbow vertex-disconnection coloring of G. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by r vd (G), is the minimum number of colors that are needed to make G rainbow.

Rainbow vertex antimagic coloring has been developed by many researchers on various kinds of graphs. For instance, in Marsidi's [8] research on the rain-bow vertex antimagic coloring of tree graphs. For the paths Pn, wheels Wn, friendships Fnm, and fans Fn in 2022, Marsidi [9] determined rainbow vertex antimagic coloring.

We will calculate the value of the rainbow vertex antimagic connection. RAINBOW INDUCED SUBGRAPHS IN PROPER VERTEX COLORINGS ANDRZEJ KISIELEWICZ AND MAREK SZYKULA that every proper vertex coloring of G contains a rainbow induced subgraph isomorphic to H. We give upper and lower bounds for ρ(H), compute the exact value for some classes of graphs, and.

Along with its variants, which consider vertex colorings and/or so-called strong colorings, the rainbow connection number has been studied from both the algorithmic and graph.