Commuting Edge-Switch Squares in Induced-Subgraph Overlaps of 42-Vertex Ramsey (5, 5) Colourings
Year: 2026
Type: preprint
Abstract: We investigate induced-subgraph overlaps within a specified catalogue of 656 (5,5)-good two-colourings of K42, represented by 328 colour-complement classes. The graph joining classes that share an induced, colour-equivalent K41 has 1,269 edges and two connected components of sizes 280 and 48. No classes in opposite components share an induced colour-equivalent K39. At order 38, precisely 262 of the 13,440 cross-component pairs share an induced colouring. We classify these overlaps constructively: 256 form sixteen edge-disjoint complete bipartite graphs K4,4 supported by commuting squares of disjoint single-edge colour switches; the six remaining pairs involve two classes related to a one-vertex replacement and a single-edge switch. The 262 positive instances are certified by explicit 38-point bijections. All negative instances are excluded by complete searches using necessary colour-invariant signatures, and the K41 and K39 computations are reproduced from the input catalogue. The participating cross-component network has eight connected components of two explicitly described types. This classification pertains only to the specified finite catalogue and neither proves its completeness nor resolves R(5,5).
Source: Preprints.org
Author: Zlatko Pangarić
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225819842
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