let $A=\lbrace a_1,\,\dots,\,a_k\rbrace $ and $B=\lbrace b_1,\,\dots,\,b_{2k}\rbrace,\ A\cap B=\emptyset$ be be a partition of a graph's vertex set $V$, i.e. $V\,=\,A\cup B$.

Question:

has $G:=\bigcup\limits_{a_i\in A}K_{a_i,B}\ \cup\ K_B$ the topology of the densest symmetric graph with weighted edges whose optimal $3D$-matching can be calculated efficiently as its lightest 2-factor?
If yes, the number $|E|$ of edges would be $\frac{4}{3}n^2-\frac{1}{3}n$

I used nonstandard notations $K_{a_i,B}$ resp. $K_B$ to denote the bipartite graphs that connect vertex $a_i\in A$ with all vertices of $B$ resp. the clique of $B$'s vertices.
The weight of the $3D$-matching is assumed to be the sum of edge weights of the $K_3$ that resemble the matching.

let $A=\lbrace a_1,\,\dots,\,a_k\rbrace $ and $B=\lbrace b_1,\,\dots,\,b_{2k}\rbrace,\ A\cap B=\emptyset$ be be a partition of a graph's vertex set $V$, i.e. $V\,=\,A\cup B$.

Question:

has $G:=\bigcup\limits_{a_i\in A}K_{a_i,B}\ \cup\ K_B$ the topology of the densest symmetric graph with weighted edges whose optimal $3D$-matching can be calculated efficiently as its lightest 2-factor?
If yes, the number $|E|$ of edges would be $\frac{4}{3}n^2-\frac{1}{3}n$

I used nonstandard notations $K_{a_i,B}$ resp. $K_B$ to denote the bipartite graphs that connect vertex $a_i\in A$ with all vertices of $B$ resp. the clique of $B$'s vertices.
The weight of the $3D$-matching is assumed to be the sum of edge weights of the $K_3$ that resemble the matching.

Addendum:
for a given complete graph with with weigthed edges and $n=3k$ vertices a heuristic for determining the set $B$ with $2k$ vertices could be to take the vertices that are adjacent to the $k$ edges of the fixed-size minimum weight matching of $2k$ vertices.
As the the vertices in $A$ are mutually not adjacent it seems reasonable to have pairs of nearby vertices in $B$

let $A=\lbrace a_1,\,\dots,\,a_k\rbrace $ and $B=\lbrace b_1,\,\dots,\,b_{2k}\rbrace,\ A\cap B=\emptyset$ be be a partition of a graph's vertex set $V$, i.e. $V\,=\,A\cup B$.

Question:

has $G:=\bigcup\limits_{a_i\in A}K_{a_i,B}\ \cup\ K_B$ the topology of the densest symmetric graph with weighted edges whose $3D$-matching can be calculated efficiently as its lightest 2-factor?
If yes, the number $|E|$ of edges would be $\frac{4}{3}n-\frac{1}{3}$

I used nonstandard notations $K_{a_i,B}$ resp. $K_B$ to denote the bipartite graphs that connect vertex $a_i\in A$ with all vertices of $B$ resp. the clique of $B$'s vertices.
The weight of the $3D$-matching is assumed to be the sum of edge weights of the $K_3$ that resemble the matching.

let $A=\lbrace a_1,\,\dots,\,a_k\rbrace $ and $B=\lbrace b_1,\,\dots,\,b_{2k}\rbrace,\ A\cap B=\emptyset$ be be a partition of a graph's vertex set $V$, i.e. $V\,=\,A\cup B$.

Question:

has $G:=\bigcup\limits_{a_i\in A}K_{a_i,B}\ \cup\ K_B$ the topology of the densest symmetric graph with weighted edges whose optimal $3D$-matching can be calculated efficiently as its lightest 2-factor?
If yes, the number $|E|$ of edges would be $\frac{4}{3}n^2-\frac{1}{3}n$

I used nonstandard notations $K_{a_i,B}$ resp. $K_B$ to denote the bipartite graphs that connect vertex $a_i\in A$ with all vertices of $B$ resp. the clique of $B$'s vertices.
The weight of the $3D$-matching is assumed to be the sum of edge weights of the $K_3$ that resemble the matching.