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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).)