Properly Colored Notions of Connectivity - A Dynamic Survey

Properly Colored Notions of Connectivity - A Dynamic Sur vey Xueliang Li 0 Nankai University 0 0 0 Georgia Southern University , USA Follow this and additional works at: https://digitalcommons.georgiasouthern.edu/tag Part of the Discrete Mathematics and Combinatorics Commons Recommended Citation - Properly Colored Notions of Connectivity - A Dynamic Survey Thi s dynamic survey is available in The ory and Applications of Graphs: https://digitalcommons.georgiasouthern.edu/tag/vol0/iss1/ 2 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. Revision History • Revision 1: December, 2016 • Original: December, 2015 If you have corrections, updates or new results which fit the scope of this work, please contact Colton Magnant at . Contents 16.4 Proper Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Minimum Spanning Subgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction An edge-colored graph is said to be properly colored if no two adjacent edges share a color. An edge-colored connected graph G is called properly connected if between every pair of distinct vertices, there exists a path that is properly colored. The proper connection number of a connected graph G, defined in [ 7 ] and also 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. If there cannot be a direct connection between two towers A and B, say for example if there is a mountain in between, there must be a route through other towers to get from A to B. As a wireless transmission passes through a signal tower, to avoid interference, it would help if the incoming signal and the outgoing signal do not share the same frequency. Suppose we assign a vertex to each signal tower, an edge between two vertices if the corresponding signal towers are directly connected by a signal and assign a color to each edge based on the assigned frequency used for the communication. Then the number of frequencies needed to assign the connections between towers so that there is always a path avoiding interference between each pair of towers is precisely the proper connection number of the corresponding graph. Aside from the above application, properly colored paths and cycles appear in a variety of other fields including genetics [ 20, 21, 22 ] and social sciences [ 17 ]. There is also a good survey [ 3 ] dealing with the case where two colors are used on the edges. More recently, there has also been another survey of the area in Chapter 16 of [ 4 ]. Unless otherwise stated, we focus on coloring edges so “coloring” will mean edge-coloring. In most cases, k will generally be used to denote the number of colors used on the edges and n will generally be used to denote the order of G, that is, the number of vertices. Also define the color degree dc(v) to be the number of colors on edges incident to v. The connectivity of a graph G, denoted by κ(G), is the minimum order of a set of vertices such that its removal results in at least two components. The chromatic number χ(G) (and similarly edge chromatic number χ′(G)) is the minimum number of colors needed to color the vertices (respectively edges) so that no two adjacent vertices (respectively edges) receive the same color. For all other standard terminology, we refer the reader to [ 14 ]. The notion of proper edge-colorings has been extremely popular since the classical work of Vizing [ 46 ]. More recently, several works have considered properly colored subgraphs as opposed to looking at the entire graph. See [ 3 ] for a survey of work concerning properly colored cycles and paths in graphs and multigraphs. The definition and study of the proper connection number was inspired by the rainbow connection number, defined by Chartrand et al. in [ 13 ]. A path is called rainbow if no two edges in the path share a color. 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. By replacing “rainbow” with “proper”, it is easy to see where the definition of the proper connection number originated. Furthermore, the strong 2 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). More generally, a graph G is said to be k-properly connected if between every pair of distinct vertices, there exist k internally disjoint properly colored paths. The k-proper connection number of a k-connected graph G, denoted by pck(G), is the minimum n (...truncated)