Rainbow Vertex-connection Number of Ladder and Mbius Ladder

A vertex-colored graph G is said to be rainbow vertex-connected if every two vertices of G are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. If for every pair u, v of distinct vertices, G contains a rainbow u-v geodesic, then G is strong rainbow vertex-connected. The minimum number k for which there exists a k-vertex-coloring of G that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of G, denoted by srvc(G). Observe that rvc(G) ≤ srvc(G) for any nontrivial connected graph G. In this paper, for a Ladder L_n,we determine the exact value of srvc(L_n) for n even. For n odd, upper and lower bounds of srvc(L_n) are obtained. We also give upper and lower bounds of the(strong) rainbow vertex-connection number of Mbius Ladder.

The Rainbow Vertex-disconnection in Graphs

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S of G such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of G-S;whereas when x and y are adjacent, S + x or S + y is rainbow and x and y belong to different components of(G-xy)-S. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertexdisconnected. In this paper, we characterize all graphs of order n with rainbow vertex-disconnection number k for k ∈ {1, 2, n}, and determine the rainbow vertex-disconnection numbers of some special graphs. Moreover, we study the extremal problems on the number of edges of a connected graph G with order n and rvd(G) = k for given integers k and n with 1 ≤ k ≤ n.

Graphs with vertex rainbow connection number two

An edge colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. A vertex colored graph G is vertex rainbow connected if any two vertices are connected by a path whose internal vertices have distinct colors. The vertex rainbow connection number of G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G vertex rainbow connected. In 2011, Kemnitz and Schiermeyer considered graphs with rc(G) = 2.We investigate graphs with rvc(G) = 2. First, we prove that rvc(G) 2 if |E(G)|≥n-22 + 2, and the bound is sharp. Denote by s(n, 2) the minimum number such that, for each graph G of order n, we have rvc(G) 2provided |E(G)|≥s(n, 2). It is proved that s(n, 2) = n-22 + 2. Next, we characterize the vertex rainbow connection numbers of graphs G with |V(G)| = n, diam(G)≥3 and clique number ω(G) = n- s for 1≤s≤4.