The concept of monochromatic connectivity was introduced by Caro and Yuster. A path in an edge-colored graph is called a monochromatic path if all the edges on the path are colored the same. An edge-coloring of G is a monochromatic connection coloring (M C-coloring, for short) if there is a monochromatic path joining any two vertices in G. The monochromatic connection number, denoted by mc(G), is defined to be the maximum number of colors used in an M C-coloring of a graph G. In this paper, we study the monochromatic connection number on the lexicographical, strong, Cartesian and direct product and present several upper and lower bounds for these products of graphs.
A vertex-coloured 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 colours, 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 colours that are needed in order to make G rainbow vertex-connected. In this paper, we study the rainbow vertexconnection number on the lexicographical, strong, Cartesian and direct product and present several upper bounds for these products of graphs. The rainbow vertex-connection number of some product networks is also investigated in this paper.
A path P in an edge-colored graph G is called a proper path if no two adjacent edges of P are colored the same, and G is proper connected if every two vertices of G are connected by a proper path in G. The proper connection number of a connected graph G, denoted by pc(G), is the minimum number of colors that are needed to make G proper connected. In this paper, we study the proper connection number on the lexicographical, strong, Cartesian, and direct product and present several upper bounds for these products of graphs.
A vertex-colored graph G is rainbow vertex-connected if two vertices are connected by a path whose internal vertices have distinct colors. 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 vertex-rainbow u−v geodesic, then G is strongly rainbow vertex-connected. The minimum k for which there exists a k-coloring of G that results in a strongly rainbow-vertex-connected graph is called the strong rainbow vertex number srvc(G) of G. Thus rvc(G) ≤ srvc(G) for every nontrivial connected graph G. A tree T in G is called a rainbow vertex tree if the internal vertices of T receive different colors. For a graph G = (V, E) and a set S ⊆ V of at least two vertices, an S-Steiner tree or a Steiner tree connecting S (or simply, an S-tree) is a such subgraph T = (V , E) of G that is a tree with S ⊆ V. For S ⊆ V (G) and |S| ≥ 2, an S-Steiner tree T is said to be a rainbow vertex S-tree if the internal vertices of T receive distinct colors. The minimum number of colors that are needed in a vertex-coloring of G such that there is a rainbow vertex S-tree for every k-set S of V (G) is called the k-rainbow vertex-index of G, denoted by rvx k (G). In this paper, we first investigate the strong rainbow vertex-connection of complementary graphs. The k-rainbow vertex-index of complementary graphs are also studied.
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