The rainbow connection number of 2-connected graphs
Jan Ekstein, P\v{r}emysl Holub, Tom\'a\v{s} Kaiser, Maria Koch,, Stephan Matos Camacho, Zden\v{e}k Ryj\'a\v{c}ek, Ingo Schiermeyer
TL;DR
This paper proves that the rainbow connection number of any 2-connected graph with n vertices is at most ceiling of n/2, improving previous bounds and establishing the optimality of this limit.
Contribution
It establishes a tight upper bound of ceiling of n/2 for the rainbow connection number of 2-connected graphs, refining earlier results.
Findings
Bound of ceiling(n/2) is optimal for 2-connected graphs.
Improves previous upper bounds on rainbow connection number.
Provides a constructive proof for the bound.
Abstract
The rainbow connection number of a graph G is the least number of colours in a (not necessarily proper) edge-colouring of G such that every two vertices are joined by a path which contains no colour twice. Improving a result of Caro et al., we prove that the rainbow connection number of every 2-connected graph with n vertices is at most the ceiling of n/2. The bound is optimal.
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph Labeling and Dimension Problems