Spanning Eulerian Subgraphs in claw-free graphs
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Bibliographic record
Abstract
A graph is claw-free if it has no induced K1,3 subgraph. A graph is essential 4-edge-connected if removing at most three edges, the resulting graph has at most one component having edges. In this note, we show that every essential 4-edgeconnected claw free graph has a spanning Eulerian subgraph with maximum degree at most 4. The graphs in this paper are finite and loopless. For terms not defined here, see Bondy and Murty [1]. Let G be a graph. Then δ(G) denotes the minimum degree of G. An edge subset X ⊆ E(G) is an essential edge-cut of G if each component of G − X has atleast an edge. A graph G is essentially k-edge-connected if for any X ⊆ E(G) with |X | < k, at most one component of G − X has edges. The line graph L(G) of a graph G has E(G) as its vertex set and two vertices of L(G) are adjacent if and only if they are adjacent as edges in G. A graph is called claw-free if it has not induced subgraph isomorphic to K1,3. It is well-known that a line graph is claw-free; and that for a graph G that is not isomorphic to a K1,n−1, G is essentially k-edge-connected if and only if L(G) is k-connected. If G has a cycle containing every vertex of G, then G is called Hamiltonian. A graph G is even if every vertex of G has even degree. A graph is Eulerian if it is connected even graph. A graph with a spanning Eulerian subgraph H with ∆(H) = 2 is Hamiltonian. A graph is supereulerian if it has a spanning Eulerian subgraph. An Eulerian subgraph H of G is dominating if E(G − V (H)) = ∅. Harary and Nash-Williams [3] showed that for a connected graph G with |E(G) | ≥ 3, L(G) is Hamiltonian if and only if G contains a dominating Eulerian subgraph. The following conjectures are well known.
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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 | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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