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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.

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.

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).)