Do such instances always admit a 3D matching?

Question

I want to know whether the following kinds of special instances of the 3D Matching problem are ``yes" instances, i.e., admit a 3D matching.

We are given 3 sets $A,B,C$ containing $m$ elements each, and $n$ tuples $\{T_i\}_{i\in [n]}$ where $T_i \in A\times B\times C$. We further know:

(i) Each element of $A\cup B \cup C$ occurs exactly in two tuples. Simple counting shows $n = 2m$.

(ii) The elements of $A\cup B \cup C$ can be partitioned into $m$ singleton elements, and $m$ pairs $(x,y)$ where $\{x,y\} \subset T_i$ for some $i\in [n]$.

I want to know if such an instance admits a 3D matching. Is there a simple counter-example?

Answer 1

How about the following counter-example?

$m=2$, $n=4$.

$A=\{a_1,a_2\}$, $B=\{b_1,b_2\}$, $C=\{c_1,c_2\}$.

$T=\{(a_1,b_1,c_1), (a_1,b_2,c_2), (a_2,b_1,c_2), (a_2,b_2,c_1)\}$.

With the partition $A\cup B\cup C = \{a_1,b_1\}\cup\{a_2,b_2\}\cup\{c_1\}\cup\{c_2\}$.