A Traceability Conjecture for Oriented Graphs
Bibliographic record
Abstract
A (di)graph $G$ of order $n$ is $k$-traceable (for some $k$, $1\leq k\leq n$) if every induced sub(di)graph of $G$ of order $k$ is traceable. It follows from Dirac's degree condition for hamiltonicity that for $k\geq2$ every $k$-traceable graph of order at least $2k-1$ is hamiltonian. The same is true for strong oriented graphs when $k=2,3,4,$ but not when $k\geq5$. However, we conjecture that for $k\geq2$ every $k$-traceable oriented graph of order at least $2k-1$ is traceable. The truth of this conjecture would imply the truth of an important special case of the Path Partition Conjecture for Oriented Graphs. In this paper we show the conjecture is true for $k \leq 5$ and for certain classes of graphs. In addition we show that every strong $k$-traceable oriented graph of order at least $6k-20$ is traceable. We also characterize those graphs for which all walkable orientations are $k$-traceable.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".