Recoloring Treewidth-2 Graphs: Polynomial Diameter via Degeneracy — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The search results are dominated by unrelated physics/CS papers; the only mathematically substantive item is a graph recoloring theorem for treewidth-2 graphs. | MATH: Jerrum's theorem: any (d+2)-coloring of a d-degenerate graph transforms via adjacent colorings; treewidth-2 graphs are K4-minor-free, and their recoloring diameter is polynomial (Bonamy et al. result on shortest transformation length). | CONNECTION: Treewidth-2 graphs are exactly partial 2-trees — their structure relates to series-parallel decompositions, which are built from series and parallel compositions, echoing the golden-ratio-free but *binary tree* recursion. No direct 0.618/1.618 ratio appears. However, the degeneracy d and coloring bound d+2 mirror the chromatic polynomial's root structure; for d=2, the bound is 4 colors, and the 4-color theorem's planar case is a special limit. | DEPTH: 3 — This is a combinatorial graph theory result with algorithmic implications (recoloring paths), but i
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- The diameter of recoloring graphs under a maximum average degree bound
- Polynomial directed treewidth forces a large bramble with congestion two
- E8 Root Vector Encoding of Genetic Codon Degeneracy and Stereoalignment — E8 Intelligence Research
- Khovanov Homology and Gauge-Theoretic Extensions of the Jones Polynomial — E8 Intelligence Research
- AI-Assisted Elliptic Curve Ranks and the Perfect Cuboid Polynomial Constraint — E8 Intelligence Research
BibTeX
@misc{caldin2026recoloring,
title = {{Recoloring Treewidth-2 Graphs: Polynomial Diameter via Degeneracy — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23587152},
url = {https://zenodo.org/records/23587152},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Recoloring Treewidth-2 Graphs: Polynomial Diameter via Degeneracy — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23587152
UR - https://zenodo.org/records/23587152
AB - FINDING: The search results are dominated by unrelated physics/CS papers; the only mathematically substantive item is a graph recoloring theorem for treewidth-2 graphs. | MATH: Jerrum's theorem: any (d+2)-coloring of a d-degenerate graph transforms via adjacent colorings; treewidth-2 graphs are K_4-minor-free, and their recoloring diameter is polynomial (Bonamy et al. result on shortest transformation length). | CONNECTION: Treewidth-2 graphs are exactly partial 2-trees — their structure relates to series-parallel decompositions, which are built from series and parallel compositions, echoing the golden-ratio-free but *binary tree* recursion. No direct 0.618/1.618 ratio appears. However, the degeneracy d and coloring bound d+2 mirror the chromatic polynomial's root structure; for d=2, the bound is 4 colors, and the 4-color theorem's planar case is a special limit. | DEPTH: 3 — This is a combinatorial graph theory result with algorithmic implications (recoloring paths), but i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -