Graph classes: a survey
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The rainbow connectivity of a graph
Networks
The rainbow connection of a graph is (at most) reciprocal to its minimum degree
Journal of Graph Theory
The parameterized complexity of some minimum label problems
Journal of Computer and System Sciences
Hardness and algorithms for rainbow connection
Journal of Combinatorial Optimization
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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).