Graph Coloring | Graph Coloring Algorithm | Math@TutorCircle.com

Graph Coloring | Graph Coloring Algorithm | Math@TutorCircle.com

Graph Coloring Algorithm

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There are many heuristic sequential techniques for coloring a graph. Given below are different graph coloring algorithms.

Greed Graph Coloring: This algorithm focuses on carefully picking the next vertex to be colored. In this, once a vertex is colored, its color never changes.
Vertices are considered to be in a specific order $v_{1}, v_{2},........, v_{n}$ and $v_{i}$ is the smallest available color not used by $v_{i}$'s neighbours.
If the vertices are ordered according to their degrees, the resulting greedy coloring uses at most max$_{i}$ min{d(x$_{i}$) + 1, i} colors, at most one more than the graph's maximum degree.
This heuristic is sometimes called the Welsh–Powell algorithm.
A coloring $F$ of the vertices $v_{0}, v_{1}$,...... ,$v_{n-1}$ of the graph $G$ is tight with respect to the given order, if
$F$($a_{i}$) $\leq$ colors (i - 1) + 1 for all i = 0, 1, ...., n - 1
This is the backtracking sequential coloring algorithm, which returns the exact value of $x(G), first developed by Brown.First Fit Algorithm: This is an easiest and fastest technique of all greedy coloring heuistics. It sequentially assigns each vertex the lowest legal color. This algorithm has the advantage of being very simple and fast and this can be implemented to run in O(n).
This algorithm simply picks a vertex from an arbitrary order.

Vertex Coloring

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Vertex coloring is a way of coloring the vertices of a graph such that no two adjacent vertices share the same color.
Edge and Face coloring can be transformed into Vertex version.

In vertex coloring, given a graph, identify how many colors are required to color its vertices in such a way that no two adjacent vertices receive the same color. The required number of colors is called the chromatic number of G and is denoted by $\chi$(G). It assigns colors to each vertex of the graph in such a way that no edge connects two identically colored vertices. vertex coloring tries to minimize the number of colors for a given graph.

A vertex coloring of a graph with k or fewer colors is known as a k-coloring. A graph having a k-coloring, $\chi$(G) = k is said to be a k-colorable graph, while a graph having chromatic number $\chi$G) = k is called a k-chromatic graph

Edge Coloring

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An edge coloring of a graph G is a coloring of the edges of G such that adjacent edges receive different colors. An edge coloring containing the smallest possible number of colors for a given graph is known as a minimum edge coloring. Whether it is possible to color the edges of a given graph using at most k different colors, for a given value of k, or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph.

Face Coloring

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Face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Faces that meet only at a vertex are allowed to be colored the same color. The (face) chromatic number of a map is the smallest number of colors that can be used to color the map subject to our rule, that faces with an edge in common get different colors.

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