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In graph theory, a path in an edge-colored graph is said to be rainbow if no color repeats on it. A graph is said to be rainbow-connected (or rainbow colored ) if there is a rainbow path between each pair of its vertices .
Graph coloring enjoys many practical applications as well as theoretical challenges. Beside the classical types of problems, different limitations can also be set on the graph, or on the way a color is assigned, or even on the color itself. It has even reached popularity with the general public in the form of the popular number puzzle Sudoku ...
Given a graph G and given a set L(v) of colors for each vertex v (called a list), a list coloring is a choice function that maps every vertex v to a color in the list L(v).As with graph coloring, a list coloring is generally assumed to be proper, meaning no two adjacent vertices receive the same color.
A proper edge-coloring does not guarantee the existence of a perfect rainbow matching. For example, consider the graph K 2,2: the complete bipartite graph on 2+2 vertices. Suppose the edges (x 1,y 1) and (x 2,y 2) are colored green, and the edges (x 1,y 2) and (x 2,y 1) are colored blue. This is a proper coloring, but there are only two perfect ...
Graph theory in sociology: Moreno Sociogram (1953). [16] Graph theory is also widely used in sociology as a way, for example, to measure actors' prestige or to explore rumor spreading, notably through the use of social network analysis software. Under the umbrella of social networks are many different types of graphs. [17]
In graph theory, a rainbow-independent set (ISR) is an independent set in a graph, in which each vertex has a different color. Formally, let G = (V, E) be a graph, and suppose vertex set V is partitioned into m subsets V 1, …, V m, called "colors". A set U of vertices is called a rainbow-independent set if it satisfies both the following ...
In graph theory, an area of mathematics, an equitable coloring is an assignment of colors to the vertices of an undirected graph, in such a way that No two adjacent vertices have the same color, and; The numbers of vertices in any two color classes differ by at most one.
Spectral graph theory is the branch of graph theory that uses spectra to analyze graphs. See also spectral expansion. split 1. A split graph is a graph whose vertices can be partitioned into a clique and an independent set. A related class of graphs, the double split graphs, are used in the proof of the strong perfect graph theorem.