PARTITION AND COHEN-MACAULAY EXTENDERS

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Problem: In the paper’s setting of finite abstract pure simplicial complexes, a partition extender for a pure $d$-complex $\Delta$ is a pure $d$-complex $\Gamma\supseteq\Delta$ such that $\Gamma$ and $(\Gamma,\Delta)$ are partitionable. The phrase “minimal” is ambiguous; the preceding question explicitly suggests minimizing the number of added faces. I analyze that natural formalization: minimize $|\Gamma\setminus\Delta|$. The same example also refutes uniqueness for inclusion-minimal extenders.

Result: The minimal partition extender need not be unique.

Let

$$\Delta=\langle 12,34,56\rangle$$

be the pure 1-dimensional complex consisting of three disjoint edges. Define

$$\Gamma_1=\langle 12,34,56,13,35\rangle,\qquad \Gamma_2=\langle 12,34,56,13,45\rangle .$$

Then $\Gamma_1\setminus\Delta=\{13,35\}$ and $\Gamma_2\setminus\Delta=\{13,45\}$, so both add exactly two faces.

Both $\Gamma_i$ are partitionable. For example,

$$\Gamma_1=[\varnothing,13]\sqcup[2,12]\sqcup[4,34]\sqcup[5,35]\sqcup[6,56],$$

and

$$\Gamma_2=[\varnothing,13]\sqcup[2,12]\sqcup[4,34]\sqcup[5,45]\sqcup[6,56].$$

The relative complexes are also partitionable:

$$(\Gamma_1,\Delta)=[13,13]\sqcup[35,35],\qquad (\Gamma_2,\Delta)=[13,13]\sqcup[45,45].$$

Thus both are partition extenders.

Minimality: $\Delta$ is a forest with three connected components. A disconnected forest with at least two components is not partitionable: if a 1-dimensional forest with $n$ vertices and $m$ edges had a partitioning, the unique interval containing $\varnothing$ covers two vertices, each other covered vertex requires an interval $[v,e]$, and hence $m\ge n-1$; but a forest with $c\ge2$ components has $m=n-c\le n-2$. Adding fewer than two faces cannot connect the three components into one tree; in particular $\Delta$, or $\Delta$ plus one edge, is still a disconnected forest. Hence no partition extender adds fewer than two faces.

Finally, $\Gamma_1$ and $\Gamma_2$ are not isomorphic: their graph degree multisets are

$$\{3,2,2,1,1,1\}\quad\text{and}\quad \{2,2,2,2,1,1\},$$

respectively. Therefore uniqueness fails, even up to simplicial isomorphism.

Verification audit: the complexes are finite, pure, 1-dimensional, contain $\Delta$, and the displayed interval decompositions satisfy the paper’s definition of partitionability. No extra hypotheses are used.

Citation: Definitions and question: Doolittle, Goeckner, Lazar, “Partition and Cohen–Macaulay Extenders,” arXiv:1911.12791. No prior source for this counterexample is asserted.