Color-Distance in Color-Connected Graphs

For a connected graph G and a positive integer k, the kth power G(k) of G is the graph with V(G(k)) = V (G) where uv is an element of E(G(k)) if the distance d(G)(u, v) between u and v is at most k. The edge coloring of G(k) defined by assigning each edge uv of G(k) the color d(G)(u, v) produces an edge-colored graph G(k) called a distance-colored graph. The distance-colored graph G(k) is (properly) color-connected if every two vertices u and v in G(k) are connected by a properly colored u - v path in G(k). The color-connection exponent cce(G) of G is defined as the minimum k for which G(k) is color-connected. It is shown that cce(G) = 2 for every non-complete connected graph C. The color-distance between two vertices u and v in G(2) is the minimum length of a properly colored u - v path in G(2). The color-eccentricity ce(v) of a vertex v in G is the color-distance between v and a vertex farthest from v in G(2). The minimum color eccentricity among the vertices of G is its color-radius and the maximum eccentricity is its color-diameter. The color-center and color-periphery are defined as expected. Results and open questions are presented on these color-distance parameters. A subgraph H in the distance-colored graph G(k) is a rainbow subgraph if no two edges of H are colored the same. The graph G(k) is rainbow-connected if every two vertices u and v in G(k) are connected by a rainbow u - v path in G(k). The rainbow-connection exponent of G is defined as the minimum k for which G(k) is rainbow-connected. It is shown that for a connected graph G of diameter d >= 2, the rainbow-connection exponent of G is the unique positive integer k such that ((k)(2)) < d <= ((k+1)(2)). Furthermore, every connected graph is a rainbow subgraph for some distance-colored graph.