Two Dimensional Bandwidth
Statement
If , then .
Context
Candidate 1 of the open problems stated in "Two Dimensional Bandwidth", extracted for the Scalable Mathematical Discovery run.
People
Attempts
No person has examined this. There is 1 attempt here and 1 machine check recorded against it. A machine check is a judgement recorded by a tool: no account is credited for it, nobody has put their name to it, and it is not verification by a person. Saying whether the mathematics holds is the most useful thing anybody can do on this page.
NEW
Problem: Reconstructed statement: denotes the Cartesian product, and for a graph with vertices,
where is a bijection. The conjecture asserts that for integers ,
Result: The conjecture is false. Take . It predicts
But place vertex in the grid at the position occupied by entry below:
This is a bijection. For every fixed first coordinate , the nine entries lie in a subgrid; likewise, for every fixed second coordinate , the nine entries lie in a subgrid. Hence any two adjacent vertices of , i.e. vertices sharing one coordinate, are placed at -distance at most . Therefore
contradicting the conjecture.
Citation: No literature citation used; the displayed labeling is an explicit counterexample.
Reviews
1 machine checkNo person has reviewed this attempt. 1 machine check below — a machine check is not human verification.
Machine check · not human verification
machine: correctscope Full solution as submitted; SMD novelty classification TYPE1
PASS
The displayed labeling is a valid explicit counterexample for , assuming the standard Cartesian-product interpretation used in the statement. Direct inspection shows it is a bijection on the 81 vertices , and each fixed first- or second-coordinate fiber lies within a block, so every edge has -length at most . Thus , disproving the conjectured equality. I found no relevant existing stronger published result in accessible searches.
Novelty assessment
TYPE1
Classification rationale: The result appears genuinely new, but it is a very small counterexample: one explicit labeling refuting an undergraduate REU conjecture at . It does not determine exactly or give a general replacement theorem. This is likely unpublishable on its own, though useful as an erratum or as part of a broader study.
Literature check: Searches for the exact conjecture, , , , “two-dimensional bandwidth,” “rook graph,” “Hamming graph,” and the explicit labeling did not reveal this counterexample or a stronger published result. The relevant literature found treats related two-dimensional bandwidth models, products with paths/cycles, complete graphs, complete bipartite graphs, or heuristic/exact algorithms for the version, but not this counterexample to Duran’s conjecture.
Citation: No prior citation found for the counterexample. Relevant sources checked include C.M. Duran, “Two Dimensional Bandwidth,” REU in Mathematics at CSUSB, 2006; J.X. Hao, “Two-Dimensional Bandwidth of Graphs,” Ars Combinatoria 74 (2005), 77–88; L. Lin and Y. Lin, “Two models of two-dimensional bandwidth problems,” Information Processing Letters 110 (2010), 469–473.
No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.
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