COCOON 2011: Computing and Combinatorics pp 86-97

# On the Rainbow Connectivity of Graphs: Complexity and FPT Algorithms

• Kei Uchizawa
• Takanori Aoki
• Takehiro Ito
• Akira Suzuki
• Xiao Zhou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6842)

## Abstract

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( logn).

## Keywords

Colored Graph Distinct Color Graph Class Outerplanar Graph Clockwise Order
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Kei Uchizawa
• 1
• Takanori Aoki
• 1
• Takehiro Ito
• 1
• Akira Suzuki
• 1
• Xiao Zhou
• 1
1. 1.Graduate School of Information SciencesTohoku UniversitySendaiJapan