The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and
Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is …

In this paper, we present the total proper connection number of strong k-quasi-transitive
digraphs with diam (D)≥ k+ 2 for even k. Next, we determine the exact values of strong total …

For a directed graph D, it's deemed rainbow connected if each arc is assigned a different
color, so that all paths from the vertex u to the vertex v are rainbow connected. Rainbow …

The (strong) proper vertex connection number spvc→(Γ),(s) pv c-number for short, is
denoted as the smallest cardinality of colors required to color the digraph Γ so that Γ is …

An edge-coloured path is\emph {rainbow} if all the edges have distinct colours. For a
connected graph $ G $, the\emph {rainbow connection number} $ rc (G) $ is the minimum …

For an edge-colored graph (resp. a vertex-colored graph), a subset of edges (resp. vertices)
is called monochromatic if all elements are assigned the same color. An edge-coloring of G …

A directed path in an edge colored digraph is said to be a rainbow path if no two edges on
this path share the same color. An edge colored digraph Γ is rainbow connected if any two …

A total-colored graph G is total rainbow connected if any two vertices are connected by a
path whose edges and inner vertices have distinct colors. A graph G is total rainbow k …

In 2008, Chartrand et al. first introduced the concept of rainbow connection. Since then the
study of rainbow connection has received considerable attention in the literature, and now it …

Given a set S⊆ V (Γ) with| S|≥ 2, a tree T is considered as a total proper S-tree or a total
proper tree connecting S if any two adjacent or incident elements of edges and V (T)\S of T …