A connection between locating colorings of certain join graphs with cycles in Kautz digraphs
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Bibliographic record
Abstract
<p>A proper <span class="math inline">\(k\)</span>-coloring <span class="math inline">\(\alpha\)</span> of a graph <span class="math inline">\(G\)</span> induces a partition <span class="math inline">\(\Pi = \{C_1, C_2, \dots, C_k\}\)</span>, where <span class="math inline">\(C_i = \{v \in V(G) \mid \alpha(v) = i\}\)</span>. The color code of a vertex <span class="math inline">\(v \in V(G)\)</span> with respect to <span class="math inline">\(\Pi\)</span> is defined as the tuple <span class="math inline">\(c_{\Pi}(v) = (d(v, C_1), d(v, C_2), \dots, d(v, C_k))\)</span>, where <span class="math inline">\(d(v, C_i)\)</span> represents the distance from <span class="math inline">\(v\)</span> to the set <span class="math inline">\(C_i\)</span>. A proper <span class="math inline">\(k\)</span>-coloring <span class="math inline">\(\alpha\)</span> is called a locating <span class="math inline">\(k\)</span>-coloring of <span class="math inline">\(G\)</span> if <span class="math inline">\(\alpha\)</span> induces a partition <span class="math inline">\(\Pi\)</span> such that for any two distinct vertices <span class="math inline">\(u, v \in V(G)\)</span>, it holds that <span class="math inline">\(c_{\Pi}(u) \neq c_{\Pi}(v)\)</span>. The locating chromatic number of <span class="math inline">\(G\)</span>, denoted <span class="math inline">\(\chi_L(G)\)</span>, is the smallest <span class="math inline">\(k\)</span> for which a locating <span class="math inline">\(k\)</span>-coloring of <span class="math inline">\(G\)</span> exists. In this paper, we establish a connection between the locating <span class="math inline">\(k\)</span>-coloring of <span class="math inline">\(C_n(1,2,\dots,t) + K_m\)</span> and the union of graphs <span class="math inline">\(\bigcup_{i=1}^p C_{n_i} + K_m\)</span>, leveraging properties of simple cycles in directed graphs. Using this connection, we determine the locating chromatic number of <span class="math inline">\(C_n(1,2,\dots,t) + K_m\)</span> for <span class="math inline">\(t = 2\)</span> and <span class="math inline">\(n \in [6, 28]\)</span>, as well as for <span class="math inline">\(t = 3\)</span> and <span class="math inline">\(n \in [8, 24]\)</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