A set S of vertices in a graph G = (V,E) is called a total k-distance dominating set if every vertex in V is within distance k of a vertex in S. A graph G is total k-distance domination-critical if γ t (G − x) < γ t (G) for any vertex x ∈ V (G). In this paper, we investigate some results on total k-distance domination-critical of graphs.

Let k be a positive integer and G = (V,E) be a connected graph of order n. A set D ⊆ V is called a k-dominating set of G if each x ∈ V (G) − D is within distance k from some vertex of D. A connected k-dominating set is a k-dominating set that induces a connected subgraph of G. The connected k-domination number of G, denoted by γ k(G), is the minimum cardinality of a connected k-dominating set. ...

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → {0, 1, 2} such that for every vertex with label 0, there is a vertex with label 2 at distance at most k from each other. The weight of a k-distance Roman dominating function f is the value ω(f) = ∑ v∈V f(v). The k-distance Roman domination number ...

Let G = (V (G), E(G)) be a nontrivial connected graph of order n with an edge-coloring c : E(G) → {1, 2, . . . , q}, q ∈ N, where adjacent edges may be colored the same. A tree T in G is a rainbow tree if no two edges of T receive the same color. For a vertex set S ⊆ V (G), a tree connecting S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such t...

For positive integers k and d such that 4 ≤ k < d and k 6= 5, we determine the maximum number of rainbow colored copies of C4 in a k-edge-coloring of the d-dimensional hypercube Qd. Interestingly, the k-edge-colorings of Qd yielding the maximum number of rainbow copies of C4 also have the property that every copy of C4 which is not rainbow is monochromatic.

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 geodesi...

An edge-coloured path is rainbow if the colours of its edges are distinct. For a positive integer k, an edge-colouring of a graph G is rainbow k-connected if any two vertices of G are connected by k internally vertex-disjoint rainbow paths. The rainbow k-connection number rck(G) is defined to be the minimum integer t such that there exists an edge-colouring of G with t colours which is rainbow ...