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I'm facing the problem of proving that given graph IS NOT Hamiltonian. As far as i know, both Ore's and Durac's theorem do not work in opposite direction. Therefore i'm left with another "hint" given on lecture, stating:

  • If $G\setminus S$ yields more than $|S|$ components, it is not Hamiltonian.

I haven't found any such subset. I've seen solutions to such problems, however containing a bridge, which isn't the case here. I do also know that obviously from every vertex of degree $2$ both of its edges must be used. However I do not know how to use this knowledge here. Could you show me how to use the tools I have or which way to follow here?

enter image description here

EDIT 1: If we start constructing given graph from the outer circle, without adding any inner edges or vertex K, we have a graph that is Hamiltonian. In this case, adding any edge doesn't change it's state. Adding a vertex of even degree, with its edges connected to nod-adjacent outer egdes (for example here K connected with B and E, not B and A) seems to make it non-Hamiltonian. Although it seems to also work here, i can't figure out a formal explanation.

PS. Thank you Brian for editing.

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  • $\begingroup$ What graph do you mean, the one in the picture? Or any graph which has all vertices of degree 3? $\endgroup$ – cubeception Dec 15 '16 at 19:59
  • $\begingroup$ The one added. Seems like the title isn't exactly concrete, but misleading instead. I will change that. $\endgroup$ – Mr_Max Dec 15 '16 at 20:02
  • $\begingroup$ Is $S$ a vertex set or an edge set? Or is it both (so you delete a vertex with its edges)? $\endgroup$ – Ian Dec 15 '16 at 20:22
  • $\begingroup$ S is a vertex set. Of course deleting it deletes also adjacent edges. $\endgroup$ – Mr_Max Dec 15 '16 at 20:25
  • $\begingroup$ Well, a simple remark is that the graph without the middle vertex is Hamiltonian. So whatever you do you don't want to delete that. $\endgroup$ – Ian Dec 15 '16 at 20:32
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The Petersen graph is well-known to be non-Hamiltonian. This graph is a spanning subgraph of the Petersen graph obtained by edge deletion. Thus it is non-Hamiltonian. enter image description here

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  • $\begingroup$ The peterson graph is connected and not a tree, and therefore contains a cycle. Are you implying that such a cycle which is a proper subgraph of the peterson graph is non-hamiltonion? $\endgroup$ – JMoravitz Dec 15 '16 at 20:36
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    $\begingroup$ This graph is obtained from the Petersen graph by deleting only edges, so if the Petersen graph is not Hamiltonian, then this graph isn't Hamiltonian. $\endgroup$ – D. Thomine Dec 15 '16 at 20:39
  • $\begingroup$ @D.Thomine Now that is a good and correct argument. Now if Laars would correct his post I could remove the downvote, because as worded it is incorrect. A proper subgraph obtained only by deleting edges is a very different concept than simply a proper subgraph. $\endgroup$ – JMoravitz Dec 15 '16 at 20:43
  • $\begingroup$ @JMoravitz, I agree, that's why I'm waiting before I can upvote this answer. I just found useful to point the correct version of the argument. $\endgroup$ – D. Thomine Dec 15 '16 at 20:44
  • $\begingroup$ Fixed. I was trying to draw a picture and get it added to this post. I should have taken more time to read what I posted. $\endgroup$ – Laars Helenius Dec 15 '16 at 20:47

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