Perfect contractility and 1-amalgam blocks

Definitions and mathematical statement

The exact source contains only "Conjecture 1-Amalgam decomposition is $C_{pc}$-preserving." It does not define the decomposition, its blocks, $C_{pc}$, or the direction of preservation. The supplied aid interprets $C_{pc}$ as the class of perfectly contractile finite simple graphs, under a standard marker-block convention; this interpretation is explicit rather than an authenticated expansion of the short sentence.

An even pair is a pair of distinct nonadjacent vertices such that every induced path between them has even length; for vertices in different components the condition is vacuous. Contracting the pair replaces it by a vertex with the union of their neighborhoods. A graph is even-contractile if successive even-pair contractions reduce it to a clique, and perfectly contractile if every induced subgraph is even-contractile. Let $\mathcal C_{pc}$ denote that class.

For the working 1-amalgam model, partition \[ V(G)=X_1\mathbin{\dot\cup}X_2\mathbin{\dot\cup}K, \qquad\varnothing\ne A_i\subseteq X_i. \] Require $K$ to be a clique complete to $A_1\cup A_2$; the edges between $X_1$ and $X_2$ are exactly all edges of $A_1\times A_2$. Edges from $K$ to the remaining side vertices may be arbitrary. The marker block $G_i$ is $G[X_i\cup K]$ with one new marker adjacent exactly to $A_i\cup K$. Any additional proper-side size requirements of the source definition are not supplied. The aid's full-preservation target in this model is \[ G\in\mathcal C_{pc}\quad\Longleftrightarrow\quad G_1,G_2\in\mathcal C_{pc}. \] It reports the forward implication for arbitrary $K$, but proves the converse only in the following cases: $K=\varnothing$ and both $A_i$ are singletons; or $A_i=X_i$ for both sides, with arbitrary $K$.

For $K=\varnothing$, the aid reports exact same-side induced-path parity correspondence. An induced path with endpoints in one side either remains there or makes a two-edge excursion through one opposite attachment vertex; replacing that vertex by the marker preserves inducedness and parity. Hence same-side even pairs correspond to those in the marker block. The correspondence survives successive side contractions not involving a marker, so every marker-free block contraction sequence lifts to the original graph. This does not claim that arbitrary marker-involving contractions lift, and it does not prove the unrestricted converse.

For full attachments $A_i=X_i$, the aid invokes a join argument; with $K=\varnothing$ and singleton attachments it reports complete preservation separately. These are structural partial results under the stated model. The source's general preservation conjecture and its missing conventions remain unresolved by the supplied summary.

Short English statement

The source gives a very short preservation conjecture without definitions. Under the aid's perfectly contractile marker-block interpretation, one implication holds generally and the converse is reported for singleton or full attachments; same-side path parity gives a further marker-free reduction.

Sources

• Local UnsolvedMath registry, permanent ID $20001148$, source code catalogue entry; aim-combinatorics-notes.json, notes[272], tag conjecture.
• AIM workshop "The Perfect Graph Conjecture," source item 1: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf. Statement status exact; the supplied verification reproduces only the one-sentence conjecture. No PDF was accessed for this draft.
• No original remarks, literature field, proposer, year, authoritative definition reference, or external citation for the proposed marker-block partial results is supplied.
• Research attempt 1: partial_result, problem status at run unknown, rigor label proved, novelty confidence low; registry status partially_solved, classification PARTIAL-PROGRESS, suggested difficulty L4. Created and updated 14 August 2026. This machine-generated aid is not peer reviewed; independently verify before citation.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0.