On strong rainbow connection number
Xueliang Li, Yuefang Sun

TL;DR
This paper studies the strong rainbow connection number of graphs, characterizing graphs with high values, providing bounds based on edge-disjoint triangles, and identifying conditions for equality.
Contribution
It characterizes graphs with strong rainbow connection number close to the number of edges and establishes bounds related to edge-disjoint triangles.
Findings
Graphs with $src(G)=m$ are trees.
Graphs with $src(G) eq m-1$ are not trees.
Characterization of graphs with $src(G)=m-2$.
Abstract
A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. For any two vertices and of , a rainbow geodesic in is a rainbow path of length , where is the distance between and . The graph is strongly rainbow connected if there exists a rainbow geodesic for any two vertices and in . The strong rainbow connection number of , denoted , is the minimum number of colors that are needed in order to make strong rainbow connected. In this paper, we first investigate the graphs with large strong rainbow connection numbers. Chartrand et al. obtained that is a tree if and only if , we will show that , so is not a tree if and only if , where is the number of edge of .…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph Labeling and Dimension Problems
