Spanning Eulerian Subgraphs in claw-free graphs
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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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 source (direct Gemma or distilled Codex), 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".