Path-systems in regular graphs and bipartite graphs
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Bibliographic record
Abstract
<p>We say that a graph <span class="math inline">\(G\)</span> has a <span><em>path-system</em></span> with respect to a set <span class="math inline">\(W\)</span> of even number of vertices in <span class="math inline">\(G\)</span> if <span class="math inline">\(G\)</span> has vertex-disjoint paths <span class="math inline">\(P_1,P_2, \ldots, P_m\)</span> such that (i) each path <span class="math inline">\(P_i\)</span> connects two vertices of <span class="math inline">\(W\)</span> and (ii) the set of end-vertices of the paths <span class="math inline">\(P_i\)</span> is exactly <span class="math inline">\(W\)</span>. In particular, <span class="math inline">\(m=|W|/2\)</span>. Moreover, if <span class="math inline">\(G\)</span> has a path-system with respect to every set <span class="math inline">\(W\)</span> of even number of vertices in <span class="math inline">\(G\)</span>, we say that <span class="math inline">\(G\)</span> has a <span><em>path system</em></span>. We prove the following theorems: (i) if <span class="math inline">\(G\)</span> is an <span class="math inline">\(r\)</span>-edge-connected <span class="math inline">\(r\)</span>-regular graph, then for any <span class="math inline">\(r-1\)</span> edges <span class="math inline">\(e_1,\ldots, e_{r-1}\)</span>, <span class="math inline">\(G-\{e_1,\ldots, e_{r-1}\}\)</span> has a path-system, (ii) every <span class="math inline">\(k\)</span>-connected <span class="math inline">\(K_{1,k+1}\)</span>-free graph has a path-system, and (iii) if a connected bipartite graph <span class="math inline">\(G\)</span> with bipartition <span class="math inline">\((A,B)\)</span> satisfies <span class="math inline">\(|A| \le 2|B|\)</span>, <span class="math inline">\(|N_G(X)| \ge 2|X|\)</span> or <span class="math inline">\(N_G(X)=B\)</span> for all <span class="math inline">\(X\subseteq A\)</span>, and <span class="math inline">\(|N_G(Y)| \ge |Y|\)</span> or <span class="math inline">\(N_G(Y)=A\)</span> for all <span class="math inline">\(Y\subseteq B\)</span>, then <span class="math inline">\(G\)</span> has a path-system with respect to every set <span class="math inline">\(W\)</span> of even number of vertices of <span class="math inline">\(A\)</span>.</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 | 0.001 | 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 | 0.000 | 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