Easy instances of the coloring problem on graphs with degree at most 4












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Given a graph with some set of colors, the goal of the coloring problem is to color the input graph with as few number of colors as possible, such that no adjacent vertices have the same color.



In general, the problem is hard. For trees and cycles, the problem is solvable in polynomial time. I am interested in bounded degree instances or more precisely graphs of degree at most 4. Is there any non-trivial subclass of graphs of degree at most 4 for which the coloring problem is solvable in polynomial time?






complexity-theory graphs colorings







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$endgroup$

















    2












    $begingroup$


    Given a graph with some set of colors, the goal of the coloring problem is to color the input graph with as few number of colors as possible, such that no adjacent vertices have the same color.



    In general, the problem is hard. For trees and cycles, the problem is solvable in polynomial time. I am interested in bounded degree instances or more precisely graphs of degree at most 4. Is there any non-trivial subclass of graphs of degree at most 4 for which the coloring problem is solvable in polynomial time?






    complexity-theory graphs colorings







    share|cite|improve this question











    $endgroup$















      2












      2








      2


      2



      $begingroup$


      Given a graph with some set of colors, the goal of the coloring problem is to color the input graph with as few number of colors as possible, such that no adjacent vertices have the same color.



      In general, the problem is hard. For trees and cycles, the problem is solvable in polynomial time. I am interested in bounded degree instances or more precisely graphs of degree at most 4. Is there any non-trivial subclass of graphs of degree at most 4 for which the coloring problem is solvable in polynomial time?






      complexity-theory graphs colorings







      share|cite|improve this question











      $endgroup$




      Given a graph with some set of colors, the goal of the coloring problem is to color the input graph with as few number of colors as possible, such that no adjacent vertices have the same color.



      In general, the problem is hard. For trees and cycles, the problem is solvable in polynomial time. I am interested in bounded degree instances or more precisely graphs of degree at most 4. Is there any non-trivial subclass of graphs of degree at most 4 for which the coloring problem is solvable in polynomial time?






      complexity-theory graphs colorings




      complexity-theory graphs colorings






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 1 hour ago









      Discrete lizard♦

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      asked 2 hours ago









      I_wil_break_wallI_wil_break_wall

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          $begingroup$

          You can of course take any graph class for which coloring is easy, and additionally require that the maximum degree is at most 4. For example, every bipartite graph of maximum degree at most 4 works. Or "bipartite" could be replaced with say "outerplanar" or more generally "small cliquewidth".



          But having maximum degree at most 4 alone will not work. That is, coloring remains hard for (i) 4-regular graphs and (ii) planar graphs of maximum degree 4.






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            $begingroup$

            You can of course take any graph class for which coloring is easy, and additionally require that the maximum degree is at most 4. For example, every bipartite graph of maximum degree at most 4 works. Or "bipartite" could be replaced with say "outerplanar" or more generally "small cliquewidth".



            But having maximum degree at most 4 alone will not work. That is, coloring remains hard for (i) 4-regular graphs and (ii) planar graphs of maximum degree 4.






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              You can of course take any graph class for which coloring is easy, and additionally require that the maximum degree is at most 4. For example, every bipartite graph of maximum degree at most 4 works. Or "bipartite" could be replaced with say "outerplanar" or more generally "small cliquewidth".



              But having maximum degree at most 4 alone will not work. That is, coloring remains hard for (i) 4-regular graphs and (ii) planar graphs of maximum degree 4.






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                You can of course take any graph class for which coloring is easy, and additionally require that the maximum degree is at most 4. For example, every bipartite graph of maximum degree at most 4 works. Or "bipartite" could be replaced with say "outerplanar" or more generally "small cliquewidth".



                But having maximum degree at most 4 alone will not work. That is, coloring remains hard for (i) 4-regular graphs and (ii) planar graphs of maximum degree 4.






                share|cite|improve this answer









                $endgroup$



                You can of course take any graph class for which coloring is easy, and additionally require that the maximum degree is at most 4. For example, every bipartite graph of maximum degree at most 4 works. Or "bipartite" could be replaced with say "outerplanar" or more generally "small cliquewidth".



                But having maximum degree at most 4 alone will not work. That is, coloring remains hard for (i) 4-regular graphs and (ii) planar graphs of maximum degree 4.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 1 hour ago









                JuhoJuho

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                15.4k54191






























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