A special perfect matching in a complete edge-colored graph Answer

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A special perfect matching in a complete edge-colored graph

In 2018 Mario Krenn posed this question, originated from recent advances in quantum physics. Despite very intensive Krenn’s promotion and our efforts, the question is answered only in special cases. For instance, the case of $n=4$ vertices was solved by kevin by means of Groebner bases. But, as far as I know, already for $n=6$ the answer is unknown and even a description of two-colored answers seems to be a too hard problem.

So now we are considering the case of the monochrome-edge-only graph. In this case a weight of the vertex-colored graph splits into a product of weights of its vertex-monochrome subgraphs, which very simplifies the problem and allows to effectively attack it by means of graph theory. This allowed us to obtain upper bounds on a number $C(n)$ of colors, which can have a monochrome-edge-only monochromatic graph on $n$ vertices. As far as I know, the present best bounds are $C(6)=2$, $2\le C(8)\le 3$and $2\le C(n)\le n-2$ for all even $n\ge 6$. Now I am trying to improve the latter bound. Namely, given an even number $n\ge 6$ and a positive integer $kwe have $C(n) provided the following claim holds.

claims. let $G=(V,E)=K_n$ be a complete graph on $n$ vertices, $(V,E_1), \dots, (V,E_{nk}) $ be spanning pairwise edge-disjoint subgraphs of $G$ without isolated vertices, and $R=E\setminus\bigcup_{i=1}^{nk} E_i$ be the set of remaining edges of $G$. Then there exists a perfect matching $M\subseteq E\set minus R$ of $G$ search that for any $i\in\{1,\dots,nk\}$, $M\not\subseteq E_i$ and a graph $(V_i, E’_i)$ has a unique perfect matching. Here $V_i$ is the set of vertices of $G$ incident to edges in $M\cap E_i$ other $E’_i$ is the set of edges of $E_i\cup R$ with both vertices in $V_i$.

In order to feel Claim, let us consider its simple cases.

for $k=1$ Claim is easy to show. Indeed, in this case $E_1,\dots, E_{nk}$ is a partition of $E$ into perfect matchings, so the set $R$ is empty. It is easy to show that $G$ has $(n-1)!!>nk$ perfect matchings, like that $G$ has a perfect matching $M$ distinct from any of $E_i$. let $i\in \{1,\dots,nk\}$ be any number. Since the set $R$ is empty and $E_i$ is a matching, the set $E’_i$ is contained in $M$so it constitutes a unique perfect matching for the graph $(V_i, E’_i)$.

But claim looks non-trivial already for $k=2$. In this case each graph $(V,E_i)$ has maximum vertex degree at most two, so it is a union of pairwise vertex-disjoint paths or cycles and $R$ is a matching, which follows that $E’_i$ is a union of at most two matchings.

Concerning the general case, given $n$I want to prove Claim for the maximum possible $k$. I think it would be nice to prove Claim for $k=\Omega(n)$. Or it would be good to know that the maximum possible $k$ is $o(n)$.

for $k=n/2+1$ there is the following simple counterexample for Claim. let $G’$ be an arbitrary subgraph of $G$ which is a copy of a complete bipartite graph $K_{n/2,n/2}$, $V’\cup V”$ be the bipartition of the vertices of $G’$and $E_1,\dots, E_{n/2}$ be an arbitrary 1 factorization of $G’$. let $M\subseteq E\setminus R=\bigcup_{i=1}^{n/2{\color{red}{-1}}} E_i $ be any perfect matching of $G$. Since $M$ consists of $n/2$ edges, by the pigeonhole principle, there exist two edges $(v’_1,v”_1)$ other $(v’_2,v”_2)$ of $M$ (with $v’_1,v’_2\in V’$ other $v”_1,v”_2\in V”$) belonging to $E_i$ for some $i\in\{1,\dots,n/2-1\}$. Then edges $(v’_1, v’_2)$ other $(v”_1, v”_2)$ belong to $R$so a graph $(V_i, E’_i)$ has a perfect matching $\{(v’_1,v’_2), (v”_1,v”_2)\}\cup M\setminus \{(v’_1,v”_1), (v’_1,v ”_1)\}$ distinct from $M$.

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