Abstract
A path in an edge-colored graph is properly colored if no two consecutive edges receive the same color. In this survey, we gather results concerning notions of graph connectivity involving properly colored paths.
Highlights
An edge-colored graph is said to be properly colored if no two adjacent edges share a color
The proper connection number of a connected graph G, defined in [7] and studied in [1] and [35], is the minimum number of colors needed to color the edges of G to make it properly connected
When building a communication network between wireless signal towers, one fundamental requirement is that the network is connected
Summary
An edge-colored graph is said to be properly colored if no two adjacent edges share a color. The proper connection number of a connected graph G, defined in [7] and studied in [1] and [35], is the minimum number of colors needed to color the edges of G to make it properly connected. The rainbow connection number of a graph G, denoted by rc(G), is the minimum number of colors needed to color the graph so that between each pair of vertices, there is a rainbow path. Rainbow connection number, denoted by src(G), is the minimum number of colors needed to color the graph so that between every pair of vertices, there is a rainbow colored geodesic (shortest path). The k-proper connection number of a k-connected graph G, denoted by pck(G), is the minimum number of colors needed to color the edges of G to make it k-properly connected.
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