# Do such instances always admit a 3D matching?

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?

$$m=2$$, $$n=4$$.
$$A=\{a_1,a_2\}$$, $$B=\{b_1,b_2\}$$, $$C=\{c_1,c_2\}$$.
$$T=\{(a_1,b_1,c_1), (a_1,b_2,c_2), (a_2,b_1,c_2), (a_2,b_2,c_1)\}$$.
With the partition $$A\cup B\cup C = \{a_1,b_1\}\cup\{a_2,b_2\}\cup\{c_1\}\cup\{c_2\}$$.