Rainbow connection number and the number of blocks
Xueliang Li, Sujuan Liu

TL;DR
This paper establishes tight upper bounds on the rainbow connection number for connected graphs with cut vertices and for bridgeless graphs, linking these bounds to the graph's block structure and connectivity.
Contribution
It introduces new tight upper bounds for the rainbow connection number based on the number of blocks with even order and connectivity properties.
Findings
For graphs with cut vertices, $rc(G) \,\leq\, (n + r - 1)/2$, where $r$ is the number of even-ordered blocks.
For 2-edge-connected (bridgeless) graphs, the paper provides a tight upper bound on $rc(G)$.
The bounds are proven to be tight, meaning they are the best possible.
Abstract
An edge-colored graph is rainbow connected if every pair of vertices of are connected by a path whose edges have distinct colors. The rainbow connection number of is defined to be the minimum integer such that there exists an edge-coloring of with colors that makes rainbow connected. For a graph without any cut vertex, i.e., a 2-connected graph, of order , it was proved that and the bound is tight. In this paper, we prove that for a connected graph of order with cut vertices, , where is the number of blocks of with even orders, and the upper bound is tight. Moreover, we also obtain a tight upper bound for a bridgeless graph, i.e., a 2-edge-connected graph.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Graph Labeling and Dimension Problems
