For a graph G = (V, E) and a color set C, let f : E → C be an edge-coloring of G which is not necessarily proper. Then, the graph G edge-colored by f is rainbow connected if every two vertices of G has a path in which all edges are assigned distinct colors. Chakraborty et al. defined the problem of determining whether the graph colored by a given edge-coloring is rainbow connected. Chen et al. introduced the vertex-coloring version of the problem as a variant, and we introduce the total-coloring version in this paper. We settle the precise computational complexities of all the three problems from two viewpoints, namely, graph diameters and certain graph classes. We also give FPT algorithms for the three problems on general graphs when parameterized by the number of colors in C; these results imply that all the three problems can be solved in polynomial time for any graph with n vertices if |C| = O(log n).
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机译：对于图G =（v，e）和颜色集c，让f：e→c是g的边缘着色，这不一定是正确的。然后，如果G的每两个顶点都有一条路径，则F F由F的图形G边彩色是彩虹连接的，其中所有边缘都被分配了不同的颜色。 Chakraborty等。确定确定由给定边缘着色的曲线图是彩虹连接的问题。陈等。将问题作为变体的顶点着色版本介绍，我们在本文中介绍了全彩色版本。从两个观点来看，我们解决了所有三个问题的精确计算复杂性，即图形直径和某些图形类。当通过C中的颜色数量参数化，我们还为一般图中的三个问题提供FPT算法;这些结果意味着，如果| C | C |将所有三个问题都可以在多项式时间内解决任何曲线图= o（log n）。
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