On (k, t)-choosability of graphs.
Bibliographic record
Abstract
A k-list assignment L of a graph G is a mapping which assigns to each vertex v of G a set L(v) of size k. A (k,t)-list assignment of G is a k-list assignment with | ⋃ v∈V (G) L(v)| = t. An L-coloring φ of G is a proper coloring of G such that φ(v) is chosen from L(v) for every vertex v. A graph G is Lcolorable if G has an L-coloring. When the parameter t is not of special interest, we simply say k-list assignment. Particularly, if L is a (k, k)-list assignment of G, then any L-coloring is called a k-coloring for G. A graph G is (k, t)-choosable if G is L-colorable for every (k, t)-list assignment L. If a graph G is (k, t)-choosable for any number t then G is k-choosable and the smallest number k satisfying this properties is called the list chromatic number of G denoted by χl(G). The list coloring problem is first studied by Vizing[6] and by Erdos, Rubin and Taylor[2]. In [2], the authors give a characterization of 2-choosable graphs. There is no literature giving a characterization of k-choosable graphs for k ≥ 3. The k-choosability of graphs is revealed only for some specific classes of graphs. For example, Thomassen[5] proves that every planar graph is 5-choosable while some planar graphs are 3-choosable. (See [4],[8],[7],[3],[9] and [10].) Ganjari et al. [1] use (k, t)-choosability of graphs to characterize uniquely 2-list colorable graphs. When k ≥ χl(G), a graph G is always (k, t)-choosable. In this paper, we focus on any integer k such that k < χl(G). For an n-vertex graph G, we find the value t in terms of n and k such that G is (k, t)-choosable. Our main study includes the following results. For fixed numbers n, k and t, every n-vertex graph is (k, t)-choosable if and only if t ≥ kn − k2 + 1. In case k ≤ t ≤ kn − k2, every n-vertex graph containing Kk+1 is not (k, t)choosable. Furthermore, every Kk+1-free n-vertex graph is (k, t)-choosable if and only if t ≥ kn−k2−2k. If k ≤ t ≤ kn−k2−2k+1, an n-vertex graph
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How this classification was reachedexpand
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 | 0.001 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".