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We have a connected graph with $2n$ nodes. Prove that exsist spanning subgraph each node with odd degree.

Idea: Let $M$ be an adjacency matrix and work all over field $\mathbb{Z}_2$. Then if $M$ is nonsingular we know that equation $$M\vec{s} = (1,1,1,....1,1)$$ has notrivial solution. Suppose set $S$ has incidence vector $\vec{s}$. Then if for each node $a$ we color each edge in $N(a)\cap S$ we win.

There are quite a few questions here. First, where did I use conectivity, second, is it $M$ really nonsingular and do we even need that and last, if all this is true does this aproach actually work?

Note: I already asked initaly question and did get an answer, so please answer just this.

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  • $\begingroup$ Sorry I am a bit confused, do you want an answer to the initial quoted question or are you only asking about your approach? $\endgroup$ – Vincenzo Dec 25 '18 at 21:34
  • $\begingroup$ @Vincenzo second one. $\endgroup$ – Aqua Dec 25 '18 at 21:35
  • $\begingroup$ I don't understand the question - clearly a simple path on $4$ vertices has an even number of vertices, but the only spanning subgraph (being the graph itself) has two vertices of even degree. $\endgroup$ – Math1000 Dec 25 '18 at 22:06
  • $\begingroup$ Take a look here. math.stackexchange.com/questions/3047513/… @Math1000 $\endgroup$ – Aqua Dec 25 '18 at 22:08
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    $\begingroup$ If you want a linear algebra solution, you should instead look at the incidence matrix of the graph and try to show that $Me={\bf 1}$ has a solution. I do not know if this makes things any easier. $\endgroup$ – Mike Earnest Dec 27 '18 at 17:44
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A base of the approach is any solution of the equation $M\vec{s} = (1,1,1,....1,1)$. But I don’t know how to show algebraically that it exists. In particular, it exists and unique provided the matrix $M$ is non-singular. Unfortunately, the adjacency matrix is singular for any graph which has two vertices with the same set of neighbors. For instance, for a cycle $C_4$ on four vertices. Also I remark that connectedness algebraically means that for each distinct indices $k,l$ there exists a power $M^k=\|m_{ij}\|$ such that $m_{i,j}>0$, but here we need to consider $M$ over $\Bbb Z$, but not over $\Bbb Z_2$.

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  • $\begingroup$ Anyway, do you have a perhaps an idea how to atack this algebraicly $\endgroup$ – Aqua Dec 25 '18 at 22:00
  • $\begingroup$ You think this is hopless try with linear algebra? $\endgroup$ – Aqua Dec 26 '18 at 22:01
  • $\begingroup$ @greedoid I rewrote my answer. Unfortunately, I don't know how to solve this problem algebraically. But I have to confess that although I have experience in graph and matrix theories, I'm ignoramus in algebraic graph theory. :-( So I put it to my ignored tags and I saw your question accidentally. $\endgroup$ – Alex Ravsky Dec 30 '18 at 16:34

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