Linear-Time and Efficient Distributed Algorithms for List Coloring\n Graphs on Surfaces
Bibliographic record
Abstract
In 1994, Thomassen proved that every planar graph is 5-list-colorable. In\n1995, Thomassen proved that every planar graph of girth at least five is\n3-list-colorable. His proofs naturally lead to quadratic-time algorithms to\nfind such colorings. Here, we provide the first such linear-time algorithms to\nfind such colorings.\n For a fixed surface S, Thomassen showed in 1997 that there exists a\nlinear-time algorithm to decide if a graph embedded in S is 5-colorable and\nsimilarly in 2003 if a graph of girth at least five embedded in S is\n3-colorable. Using the theory of hyperbolic families, the author and Thomas\nshowed such algorithms exist for list-colorings. Dvorak and Kawarabayashi\nactually gave an $O(n^{O(g+1)})$-time algorithm to find such colorings (if they\nexist) in n-vertex graphs where g is the Euler genus of the surface. Here we\nprovide the first such algorithm whose exponent does not depend on the genus;\nindeed, we provide a linear-time algorithm.\n In 1988, Goldberg, Plotkin and Shannon provided a deterministic distributed\nalgorithm for 7-coloring n-vertex planar graphs in $O(\\log n)$ rounds. In 2018,\nAboulker, Bonamy, Bousquet, and Esperet provided a deterministic distributed\nalgorithm for 6-coloring n-vertex planar graphs in $O(\\log^3 n)$ rounds. Their\nalgorithm in fact works for 6-list-coloring. They also provided an $O(\\log^3\nn)$-round algorithm for 4-list-coloring triangle-free planar graphs. Chechik\nand Mukhtar independently obtained such algorithms for ordinary coloring in\n$O(\\log n)$ rounds, which is best possible in terms of running time. Here we\nprovide the first polylogarithmic deterministic distributed algorithms for\n5-coloring n-vertex planar graphs and similarly for 3-coloring planar graphs of\ngirth at least five. Indeed, these algorithms run in $O(\\log n)$ rounds, work\nalso for list-colorings, and even work on a fixed surface (assuming such a\ncoloring exists).\n
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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.005 | 0.004 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.011 | 0.004 |
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".