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?










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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.
    34 mins ago










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















up vote
1
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?










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.
    34 mins ago










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













up vote
1
down vote

favorite









up vote
1
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?










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






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share|cite|improve this question










asked 1 hour ago









Mr. Sigma.

348116




348116












  • 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.
    34 mins ago










  • Oh, apologies. I read the title as whether minimum spanning tree have common edges.
    – Gokul
    28 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.
    34 mins ago










  • Oh, apologies. I read the title as whether minimum spanning tree have common edges.
    – Gokul
    28 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.
34 mins ago




@Gokul minimum spanning tree? What?...
– Mr. Sigma.
34 mins ago












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




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










2 Answers
2






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oldest

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2
down vote













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.






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    up vote
    2
    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



















    • 1




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











    Your Answer





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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    2
    down vote













    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
      2
      down vote













      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
        2
        down vote










        up vote
        2
        down vote









        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 26 mins ago









        Gokul

        241110




        241110






















            up vote
            2
            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



















            • 1




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















            up vote
            2
            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



















            • 1




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













            up vote
            2
            down vote










            up vote
            2
            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 22 mins ago

























            answered 28 mins ago









            Bjørn Kjos-Hanssen

            23417




            23417








            • 1




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














            • 1




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








            1




            1




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




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


















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