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Misra & Gries (1992) describe a polynomial time algorithm for coloring the edges of any graph with Δ + 1 colors, where Δ is the maximum degree of the graph. That is, the algorithm uses the optimal number of colors for graphs of class two, and uses at most one more color than necessary for all graphs.
This is a wheel graph and will therefore be optimally colored by the DSatur algorithm. Executing the algorithm results in the vertices being selected and colored as follows. (In this example, where ties occur in both of DSatur's heuristics, the vertex with lowest lexicographic labelling among these is chosen.) Vertex (color 1)
An edge coloring of a graph is a proper coloring of the edges, meaning an assignment of colors to edges so that no vertex is incident to two edges of the same color. An edge coloring with k colors is called a k -edge-coloring and is equivalent to the problem of partitioning the edge set into k matchings .
The Misra & Gries edge coloring algorithm is a polynomial time algorithm in graph theory that finds an edge coloring of any simple graph. The coloring produced uses at most Δ + 1 {\displaystyle \Delta +1} colors, where Δ {\displaystyle \Delta } is the maximum degree of the graph.
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.
Exact coloring of the complete graph K 6. Every n-vertex complete graph K n has an exact coloring with n colors, obtained by giving each vertex a distinct color. Every graph with an n-color exact coloring may be obtained as a detachment of a complete graph, a graph obtained from the complete graph by splitting each vertex into an independent set and reconnecting each edge incident to the ...
The path graph with four vertices provides the simplest example of a graph whose chromatic number differs from its Grundy number. This graph can be colored with two colors, but its Grundy number is three: if the two endpoints of the path are colored first, the greedy coloring algorithm will use three colors for the whole graph.