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| In [[graph theory]], '''path coloring''' usually refers to one of two problems:
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| * The problem of coloring a [[multiset|(multi)set]] of [[path (graph theory)|paths]] <math>R</math> in graph <math>G</math>, in such a way that any two paths of <math>R</math> which share an edge in <math>G</math> receive different colors. Set <math>R</math> and graph <math>G</math> are provided at input. This formulation is equivalent to [[Graph coloring|vertex coloring]] the ''conflict graph'' of set <math>R</math>, i.e. a graph with vertex set <math>R</math> and edges connecting all pairs of paths of <math>R</math> which are not edge-disjoint with respect to <math>G</math>.
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| * The problem of coloring (in accordance with the above definition) any chosen [[multiset|(multi)set]] <math>R</math> of paths in <math>G</math>, such that the set of pairs of end-vertices of paths from <math>R</math> is equal to some set or multiset <math>I</math>, called a ''set of requests''. Set <math>I</math> and graph <math>G</math> are provided at input. This problem is a special case of a more general class of graph routing problems, known as [[call scheduling]].
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| In both the above problems, the goal is usually to minimise the number of colors used in the coloring. In different variants of path coloring, <math>G</math> may be a [[simple graph]], [[directed graph|digraph]] or [[multigraph]].
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| ==References==
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| *[http://citeseer.ist.psu.edu/erlebach00complexity.html] ''The Complexity of Path Coloring and Call Scheduling'' by Thomas Erlebach and Klaus Jansen
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| *[http://www.nada.kth.se/~viggo/wwwcompendium/node122.html] ''A compendium of NP optimization problems'' by Viggo Kann (problem: Minimum Path Coloring)
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| [[Category:Graph coloring]]
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| {{Combin-stub}}
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Revision as of 17:38, 2 March 2014
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