Given are $n\geq 1$ permutations of $abcd$. We construct a bipartite graph $G_{a,b}$ as follows: The $n$ vertices on one side are labeled with the sets containing $a$ and the letters after it in each permutation, and the $n$ vertices on the other side are labeled with $b$ and the letters before it in each permutation. There is an edge between two vertices if their sets overlap. Construct $G_{b,c},G_{c,d},G_{d,a}$ similarly.

It could be that $G_{a,b},G_{b,c},G_{c,d}$ all have no perfect matchings (for example, if all permutations are $dcba$, then all three graphs have no edges). But is it true that among $G_{a,b},G_{b,c},G_{c,d},G_{d,a}$, at least one must have a perfect matching?

Example: if the permutation set is $\{abcd,abcd,dbca\}$, then $G_{a,b}$ has vertices $v_1 =\{a,b,c,d\},v_2 =\{a,b,c,d\},v_3 =\{a\}$ and $w_1 =\{b,a\},w_2 =\{b,a\},w_3 =\{d,b\}$ with edges $(v_1 ,w_1),(v_1 ,w_2),(v_1 ,w_3),(v_2 ,w_1),(v_2 ,w_2),(v_2 ,w_3),(v_3 ,w_1 ),(v_3 ,w_2)$.

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    $\begingroup$ Just to check for an explicit example: if the permutation set is $\{abcd,abcd,dbca\}$, then $G_{a,b}$ has vertices $v_1 = \{a,b,c,d\}, v_2 = \{a,b,c,d\}, v_3 = \{a\}$ and $w_1 = \{b,a\}, w_2 = \{b,a\}, w_3 = \{d,b\}$ with edges $(v_1,w_1), (v_1,w_2), (v_1,w_3), (v_2,w_1), (v_2,w_2), (v_2,w_3), (v_3,w_1), (v_3,w_2)$? $\endgroup$ – user113102 May 16 at 20:26
  • $\begingroup$ May I suggest that, if the explicit example above is actually correct, then it be incorporated into the main problem stmt as example? :) $\endgroup$ – antkam May 16 at 21:48
  • $\begingroup$ The example is correct, and I have added it :) $\endgroup$ – pi66 May 17 at 7:41
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    $\begingroup$ Have brute force checked it for $n \leq 8$. $\endgroup$ – user113102 May 17 at 17:55
  • $\begingroup$ I suspect if you look at the number of transpositions needed to transform one permutation into another, it will tell you a lower bound for how many of the $G_{x,y}$ the two permutations can be paired up in. $\endgroup$ – user326210 May 22 at 7:58

Not a full answer, but too long for a comment. Here's a compact representation of the graphs:

enter image description here

Each permutation is paired up with each other permutation. The entry in the table records whether there is an edge from the permutation on the left to the permutation on the top in the graph $G_{a,b}$, $G_{b,c}$, $G_{c,d}$, and $G_{d,a}$. For shorthand, these graphs are denoted by the presence or absence of a single letter, e.g. if there is an edge in $G_{a,b}$, I write $a$ in the entry. Same for $G_{b,c}$ and $b$, $G_{c,d}$ and $c$ etc.

I figure such a table may be helpful for visualization and determining an answer to this question.


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