Dominating Hadwiger's Conjecture for graphs $G$ with $α(G)=2$
Bibliographic record
Abstract
Hadwiger's Conjecture from 1943 states that every graph with chromatic number $t$ contains a $K_t$ minor. Illingworth and Wood [arXiv:2405.14299] introduced the concept of a ``dominating $K_t$ minor'' and asked whether every graph with chromatic number $t$ contains a dominating $K_t$ minor. This question is a substantial strengthening of Hadwiger's Conjecture. Norin referred to it as the ``Dominating Hadwiger's Conjecture'' and believes it is likely false. In this paper we first observe that a $t$-chromatic $G$ on $n$ vertices with independence number $α(G)\le2$ contains a dominating $K_t$ minor if and only if $G$ contains a dominating $K_{\lceil n/2\rceil}$ minor. Building on this and using a deep result of Chudnovsky and Seymour on packing seagulls, we prove that every graph $G$ on $n$ vertices with $α(G)\le 2$ and $2ω(G)\ge \lceil n/2\rceil+1$ satisfies the Dominating Hadwiger's Conjecture, where $ω(G)$ denotes the clique number of $G$. We further prove that every $H$-free graph $G$ with $α(G)\le 2$ satisfies the Dominating Hadwiger's Conjecture, where $H\in\{2K_1+P_4, K_2+2K_2, K_2+(K_1\cup K_3), K_1+(K_1\cup K_5), W_5^<, W_5^-, W_5, K_7^<, K_7^-, K_7\}$, or $H\ne K_2\cup K_3$ is any graph on at most five vertices such that $α(H)\le2$.
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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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.009 | 0.001 |
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".