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Two vertices $u, v$ of a finite graph $G(V, E)$ are said to be entangled if for any proper coloring $c:V(G)\rightarrow\mathbb{N}$ with $\chi(G)$ colors we have $c(u) = c(v)$, that is, they must have the same color.

In that question I made a false conjecture about the connectivity of entangled vertices. There I ask if

"Given a graph $G$ and two entangled vertices $u, v\in V(G)$, is there $w\in V(G)$ (possibly equal to $v$) also entangled with $u$ so that there are $\chi(G)-1$ disjoint paths from $u$ to $w$?"

It turns out that the conjecture is false for $\chi(G) \ge 5$, as shown by a counter example in that post. I would like to know now if the conjecture is true for the case $\chi(G) = 4$, i.e., if

"Given a $4$-chromatic graph $G$ and two entangled vertices $u, v\in V(G)$, is there $w\in V(G)$ (possibly equal to $v$) also entangled with $u$ so that there are $3$ disjoint paths from $u$ to $w$?"

In fact, it was this particular case that inspired me to come up with this conjecture. Any help would be greatly apreciated.

RobPratt
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Alma Arjuna
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    If you consider the complete graph $K_5$ but you remove an edge between $u$ and $v$, then $\chi(G)=4$ (because you can use the same color for $u$ and $v$) the vertices $u$ and $v$ are entangled (because they are linked to the three same vertices) and you have 3 paths going from $u$ to $v$ (one for each of the other vertices). It's just an example, but maybe we can work our way starting from this. – Fabien Sep 25 '20 at 17:29
  • Does disjoint mean edge disjoint or vertex disjoint? – N. S. Sep 25 '20 at 17:53
  • "disjoint" mean "vertex disjoint". – Alma Arjuna Sep 25 '20 at 18:18
  • I have thoroughly enjoyed working on this problem. If you intend to work more on it or publish anything, I would be keen to collaborate. – Brandon du Preez Sep 30 '20 at 23:40
  • @BrandonduPreez, I'm not sure about publishing anything. I just realized that the result I needed is slightly different from what I posted here. I intend to continue working on it, and if I get any interesting results, I will contact you again! – Alma Arjuna Oct 09 '20 at 18:53
  • @Arjuna196 Awesome! I look forward to it if you do ^^ – Brandon du Preez Oct 09 '20 at 19:02

1 Answers1

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The conjecture is true for $\chi(G) = 4$.

I have posted an answer on math overflow, which feels like a more appropriate place for the rather involved proof.

Brandon du Preez
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