An extraction fragment and an exact triangle-free list-coloring corollary

Definitions and mathematical statement

The exact canonical text, "Conjecture 2. Let $G$ be an even-hole-free graph.", contains no asserted conclusion. The supplied aid reports that it is an extraction splice: the final words of a sentence asserting $\text{Conjecture 4}\Rightarrow\text{Conjecture 3}\Rightarrow\text{Conjecture 2}$ were joined to the opening sentence of the ensuing proof. It therefore supplies no independent Conjecture 2, and the absent conjectures are not reconstructed here.

A finite simple graph is even-hole-free if it has no induced cycle of even length at least four, and triangle-free if it has no three pairwise adjacent vertices. It is 2-degenerate if every nonempty induced subgraph has a vertex of degree at most two. Write $\chi(G)$ for chromatic number and $\operatorname{ch}(G)$ for list-chromatic number: the least nonnegative integer $k$ such that every assignment of lists $L(v)$ with $|L(v)|\geq k$ admits a proper coloring $c(v)\in L(v)$. For the graph with no vertices use both numbers zero.

Separately from the invalid fragment, the aid reports this motivated corollary: every finite triangle-free even-hole-free graph $G$ is 2-degenerate and satisfies \[ \chi(G)=\operatorname{ch}(G)= \begin{cases} 0,&V(G)=\varnothing,\\ 1,&V(G)\ne\varnothing,\ E(G)=\varnothing,\\ 2,&G\text{ is a forest with at least one edge},\\ 3,&G\text{ contains a cycle}. \end{cases} \] In particular, within this triangle-free even-hole-free class, the bipartite graphs are exactly the forests. This is a separate supplied research-aid assertion, not the missing conclusion of the canonical fragment.

Short English statement

The recorded sentence is an extraction fragment rather than a conjecture. The aid separately reports an exact ordinary and list-coloring classification for finite triangle-free even-hole-free graphs, including the empty and edgeless boundary cases.

Sources

• Local registry: permanent UnsolvedMath ID $20001154$, exact public code catalogue entry; aim-combinatorics-notes.json, notes[278].
• AIM workshop "The Perfect Graph Conjecture", source item 2: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf. The exact extracted sentence is "Conjecture 2. Let G be an even-hole-free graph." No proposer, year, remarks, or literature citation is supplied.
• Research attempt 1: context_only, status at run not_a_problem, rigor label proved, novelty confidence low, suggested difficulty L2, classification OPEN-TRIAGE. It reports the source-splice diagnosis and the separate triangle-free list-coloring corollary. No authoritative separate citation for that corollary is supplied.