Facial entire coloring of 4-minor-free graphs
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Bibliographic record
Abstract
<p>Let <span class="math inline">\(G\)</span> be a plane graph. If two edges are adjacent and consecutive on the boundary walk of a face of <span class="math inline">\(G\)</span>, then they are said to be facially adjacent. We call <span class="math inline">\(G\)</span> facially entire <span class="math inline">\(k\)</span>-colorable if there is a mapping from <span class="math inline">\(V(G)\cup E(G)\cup F(G)\)</span> to a <span class="math inline">\(k\)</span> color set so that any two facially adjacent edges, adjacent vertices, adjacent faces, and incident elements receive different colors. The facial entire chromatic number of <span class="math inline">\(G\)</span> is defined to be the smallest integer <span class="math inline">\(k\)</span> such that <span class="math inline">\(G\)</span> is facially entire <span class="math inline">\(k\)</span>-colorable. In 2016, Fabrici, Jendrol’ and Vrbjarová conjectured that every connected, loopless, bridgeless plane graph is facially entire <span class="math inline">\(7\)</span>-colorable. In this paper, we give a positive answer to this conjecture for <span class="math inline">\(K_4\)</span>-minor-free graphs. More specifically, we shall prove that every <span class="math inline">\(K_{4}\)</span>-minor-free graph is facially entire <span class="math inline">\(7\)</span>-colorable.</p>
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it