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In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring [1] is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but ...
Decentralized algorithms are ones where no message passing is allowed (in contrast to distributed algorithms where local message passing takes places), and efficient decentralized algorithms exist that will color a graph if a proper coloring exists. These assume that a vertex is able to sense whether any of its neighbors are using the same ...
The contraction of the term "degree of saturation" forms the name of the algorithm. [2] DSatur is a heuristic graph colouring algorithm, yet produces exact results for bipartite, [1] cycle, and wheel graphs. [2] DSatur has also been referred to as saturation LF in the literature. [3]
By applying exact algorithms for vertex coloring to the line graph of the input graph, it is possible to optimally edge-color any graph with m edges, regardless of the number of colors needed, in time 2 m m O(1) and exponential space, or in time O(2.2461 m) and only polynomial space (Björklund, Husfeldt & Koivisto 2009).
Similarly, Liu & Zhang (2006) used the construction, starting with the Grötzsch graph, to generate many 4-critical triangle-free graphs, which they showed to be difficult to color using traditional backtracking algorithms. In polyhedral combinatorics, Euler (2003) used the Hajós construction to generate facets of the stable set polytope.
Backtracking is a class of algorithms for finding solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons a candidate ("backtracks") as soon as it determines that the candidate cannot possibly be completed to a valid solution.
A graph G is said to be perfectly orderable if there is a sequence of its vertices with the property that, for any induced subgraph of G, a greedy coloring algorithm that colors the vertices in the induced sequence ordering is guaranteed to produce an optimal coloring. For a chordal graph, a perfect elimination ordering is a perfect ordering ...
It will then proceed down the new path as it had before, backtracking as it encounters dead-ends, and ending only when the algorithm has backtracked past the original "root" vertex from the very first step. DFS is the basis for many graph-related algorithms, including topological sorts and planarity testing.