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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 connection number refers to how many colors are needed in D to achieve rainbow connectivity. Among ordered vertex pair (u,v) with arc coloring, if both arcs and internal vertices exhibit unique colors, it is referred to as a total colored graph. As the total rainbow connection number indicates, the smallest number of colors are required to total-color D. This paper focuses on investigating the rainbow connection number of the biorientation of a connected graph, as well as the total rainbow connection number.

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 connection number refers to how many colors are needed in D to achieve rainbow connectivity. Among ordered vertex pair ( u , v ) with arc coloring, if both arcs and internal vertices exhibit unique colors, it is referred to as a total colored graph. As the total rainbow connection number indicates, the smallest number of colors are required to total-color D . This paper focuses on investigating the rainbow connection number of the biorientation of a connected graph, as well as the total rainbow connection number., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Author(s).)

In a graph H with a total coloring, a path Q is a total rainbow if all elements in V(Q)∪E(Q), except for its end vertices, are assigned different colors. The total coloring of a graph H is a total rainbow connected coloring if, for any x,y∈V(H), there is a total rainbow path joining them. The total rainbow connection number trc(H) of H is the minimum integer r such that there is a total rainbow-connected coloring of H using r colors. In this paper, we study the total rainbow connection number of several graph operations (specifically, adding an edge, deleting an edge, and the Cartesian product) for which the total rainbow connection number is upper-bounded by a linear function of its radius.

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 becomes an active topic in graph theory. As a natural generalization, Uchizawa et al. (2013) and Liu et al. (2014) presented the concept of total rainbow connection, respectively. In this paper, we investigate the total rainbow connection numbers of outerplanar graphs with diameter 2. Applying our result, we improve the main result of [X. Huang, X. Li, Y. Shi, J. Yue, Y. Zhao, Rainbow connections for outerplanar graphs with diameter 2 and 3, Applied Mathematics and Computation, 242 (2014), 277–280]. Next, we revise the main result of [Y. Liu, Z. Wang, Rainbow Connection Number of the Thorn Graph, Applied Mathematical Sciences, 8 (2014), 6373–6377], and determine the total rainbow connection numbers of graphs G , where G are the thorn graph of complete graph K n * , the thorn graph of the cycle C n *. At last, we study the rainbow 2-connection numbers of some special graphs. [ABSTRACT FROM AUTHOR]

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 -connected if there is a total-coloring of G with k colors such that G is total rainbow connected. The total rainbow connection number, denoted by trc ( G ), of a graph G is the smallest k to make G total rainbow k -connected. For n, k ≥ 1, define h ( n, k ) to be the minimum size of a total rainbow k -connected graph G of order n . In this paper, we prove a sharp upper bound for trc ( G ) in terms of the number of vertex-disjoint cycles of G . We also compute exact values and upper bounds for h ( n, k ). [ABSTRACT FROM AUTHOR]

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 a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow $k$-connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study.

A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc( G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we investigate graphs with small total rainbow connection number. First, for a connected graph G, we prove that $${\text{trc(G) = 3 if}}\left( {\begin{array}{*{20}{c}} {n - 1} \\ 2 \end{array}} \right) + 1 \leqslant \left| {{\text{E(G)}}} \right| \leqslant \left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right) - 1$$ , and $${\text{trc(G)}} \leqslant {\text{6 if }}\left| {{\text{E(G)}}} \right| \geqslant \left( {\begin{array}{*{20}{c}} {n - 2} \\ 2 \end{array}} \right) + 2$$ . Next, we investigate the total rainbow connection numbers of graphs G with | V( G)| = n, diam( G) ≥ 2, and clique number ω( G) = n − s for 1 ≤ s ≤ 3. In this paper, we find Theorem 3 of [Discuss. Math. Graph Theory, 2011, 31(2): 313-320] is not completely correct, and we provide a complete result for this theorem. [ABSTRACT FROM AUTHOR]