## Abstract

For a graph G and c_{R}:E(G)→[k], a path P between u,v∈V(G) is a rainbow path if for distinct e,e^{′}∈E(P), we have c_{R}(e)≠c_{R}(e^{′}). RAINBOW k-COLORING takes a graph G and the objective is to check if there is c_{R}:E(G)→[k] such that for all u,v∈V(G) there is a rainbow path between u and v. Two variants of the above problem are SUBSET RAINBOW k-COLORING and STEINER RAINBOW k-COLORING, where we are additionally given a subset S⊆V(G)×V(G) and S^{′}⊆V(G), respectively. Moreover, the objective is to check if there is c_{R}:E(G)→[k], such that there is a rainbow path for each (u,v)∈S and u,v∈S^{′}, respectively. Under ETH, we obtain that for each k≥3: 1. RAINBOW k-COLORING has no 2^{o(|E(G)|)}n^{O(1)}-time algorithm. 2. STEINER RAINBOW k-COLORING has no 2^{o(|S|2)}n^{O(1)}-time algorithm. We also obtain that SUBSET RAINBOW k-COLORING and STEINER RAINBOW k-COLORING admit 2^{O(|S|)}n^{O(1)}- and 2^{O(|S|2)}n^{O(1)}-time algorithms, respectively.

Original language | English |
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Pages (from-to) | 140-158 |

Number of pages | 19 |

Journal | Journal of Computer and System Sciences |

Volume | 124 |

DOIs | |

State | Published - 1 Mar 2022 |

Externally published | Yes |

## Keywords

- ETH
- Fine-grained complexity
- Lower bound
- Rainbow coloring

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Networks and Communications
- Computational Theory and Mathematics
- Applied Mathematics