Do any two spanning trees of a simple graph always have some common edges?











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I tried few cases and found any two spanning tree of a simple graph has some common edges. I mean I couldn't find any counter example so far. But I couldn't prove or disprove this either. How to prove or disprove this conjecture?






graphs graph-theory spanning-trees







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  • Assuming the weights of the graph are distinct, the edge with the minimum weight will be present in all the minimum spanning trees.
    – Gokul
    1 hour ago










  • @Gokul minimum spanning tree? What?...
    – Mr. Sigma.
    1 hour ago










  • Oh, apologies. I read the title as whether minimum spanning tree have common edges.
    – Gokul
    53 mins ago















up vote
2
down vote

favorite












I tried few cases and found any two spanning tree of a simple graph has some common edges. I mean I couldn't find any counter example so far. But I couldn't prove or disprove this either. How to prove or disprove this conjecture?






graphs graph-theory spanning-trees







share|cite|improve this question






















  • Assuming the weights of the graph are distinct, the edge with the minimum weight will be present in all the minimum spanning trees.
    – Gokul
    1 hour ago










  • @Gokul minimum spanning tree? What?...
    – Mr. Sigma.
    1 hour ago










  • Oh, apologies. I read the title as whether minimum spanning tree have common edges.
    – Gokul
    53 mins ago













up vote
2
down vote

favorite









up vote
2
down vote

favorite











I tried few cases and found any two spanning tree of a simple graph has some common edges. I mean I couldn't find any counter example so far. But I couldn't prove or disprove this either. How to prove or disprove this conjecture?






graphs graph-theory spanning-trees







share|cite|improve this question













I tried few cases and found any two spanning tree of a simple graph has some common edges. I mean I couldn't find any counter example so far. But I couldn't prove or disprove this either. How to prove or disprove this conjecture?






graphs graph-theory spanning-trees




graphs graph-theory spanning-trees






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 2 hours ago









Mr. Sigma.

355116




355116












  • Assuming the weights of the graph are distinct, the edge with the minimum weight will be present in all the minimum spanning trees.
    – Gokul
    1 hour ago










  • @Gokul minimum spanning tree? What?...
    – Mr. Sigma.
    1 hour ago










  • Oh, apologies. I read the title as whether minimum spanning tree have common edges.
    – Gokul
    53 mins ago


















  • Assuming the weights of the graph are distinct, the edge with the minimum weight will be present in all the minimum spanning trees.
    – Gokul
    1 hour ago










  • @Gokul minimum spanning tree? What?...
    – Mr. Sigma.
    1 hour ago










  • Oh, apologies. I read the title as whether minimum spanning tree have common edges.
    – Gokul
    53 mins ago
















Assuming the weights of the graph are distinct, the edge with the minimum weight will be present in all the minimum spanning trees.
– Gokul
1 hour ago




Assuming the weights of the graph are distinct, the edge with the minimum weight will be present in all the minimum spanning trees.
– Gokul
1 hour ago












@Gokul minimum spanning tree? What?...
– Mr. Sigma.
1 hour ago




@Gokul minimum spanning tree? What?...
– Mr. Sigma.
1 hour ago












Oh, apologies. I read the title as whether minimum spanning tree have common edges.
– Gokul
53 mins ago




Oh, apologies. I read the title as whether minimum spanning tree have common edges.
– Gokul
53 mins ago










2 Answers
2






active

oldest

votes

















up vote
5
down vote



accepted










No, it's not true that any two spanning trees of a graph have common edges.



Consider the wheel graph:



enter image description here



You can make a spanning tree with edges "inside" the loop and another one from the outer loop.






share|cite|improve this answer




























    up vote
    3
    down vote













    No, consider the complete graph $K_4$:



    It has the following edge-disjoint spanning trees:
    enter image description here






    share|cite|improve this answer



















    • 2




      Oh! Elegant. Why I couldn't come upon this solution. ':O.
      – Mr. Sigma.
      29 mins ago











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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    5
    down vote



    accepted










    No, it's not true that any two spanning trees of a graph have common edges.



    Consider the wheel graph:



    enter image description here



    You can make a spanning tree with edges "inside" the loop and another one from the outer loop.






    share|cite|improve this answer

























      up vote
      5
      down vote



      accepted










      No, it's not true that any two spanning trees of a graph have common edges.



      Consider the wheel graph:



      enter image description here



      You can make a spanning tree with edges "inside" the loop and another one from the outer loop.






      share|cite|improve this answer























        up vote
        5
        down vote



        accepted







        up vote
        5
        down vote



        accepted






        No, it's not true that any two spanning trees of a graph have common edges.



        Consider the wheel graph:



        enter image description here



        You can make a spanning tree with edges "inside" the loop and another one from the outer loop.






        share|cite|improve this answer












        No, it's not true that any two spanning trees of a graph have common edges.



        Consider the wheel graph:



        enter image description here



        You can make a spanning tree with edges "inside" the loop and another one from the outer loop.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 51 mins ago









        Gokul

        286110




        286110






















            up vote
            3
            down vote













            No, consider the complete graph $K_4$:



            It has the following edge-disjoint spanning trees:
            enter image description here






            share|cite|improve this answer



















            • 2




              Oh! Elegant. Why I couldn't come upon this solution. ':O.
              – Mr. Sigma.
              29 mins ago















            up vote
            3
            down vote













            No, consider the complete graph $K_4$:



            It has the following edge-disjoint spanning trees:
            enter image description here






            share|cite|improve this answer



















            • 2




              Oh! Elegant. Why I couldn't come upon this solution. ':O.
              – Mr. Sigma.
              29 mins ago













            up vote
            3
            down vote










            up vote
            3
            down vote









            No, consider the complete graph $K_4$:



            It has the following edge-disjoint spanning trees:
            enter image description here






            share|cite|improve this answer














            No, consider the complete graph $K_4$:



            It has the following edge-disjoint spanning trees:
            enter image description here







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited 48 mins ago

























            answered 53 mins ago









            Bjørn Kjos-Hanssen

            24417




            24417








            • 2




              Oh! Elegant. Why I couldn't come upon this solution. ':O.
              – Mr. Sigma.
              29 mins ago














            • 2




              Oh! Elegant. Why I couldn't come upon this solution. ':O.
              – Mr. Sigma.
              29 mins ago








            2




            2




            Oh! Elegant. Why I couldn't come upon this solution. ':O.
            – Mr. Sigma.
            29 mins ago




            Oh! Elegant. Why I couldn't come upon this solution. ':O.
            – Mr. Sigma.
            29 mins ago


















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