Graph decompositions in projective geometries
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Statement
Given a difference set in a group , and given a graph of order and size , we ask whether it is possibile to label the vertices of with the elements of is such a way that every non-identity element of may be expressed as a difference of two “adjacent labels”. We conjecture that the answer is always affirmative if is regular and connected.
Context
Candidate 1 of the open problems stated in "Graph decompositions in projective geometries", extracted for the Scalable Mathematical Discovery run.
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NEW
Problem: Reconstructed statement: Let be a difference set in a finite group . If is a connected regular simple graph with vertices and edges, then there is a bijection such that
as a multiset.
This is exactly the “adjacent labels” formulation in the cited introduction; since , “every nonidentity occurs” means “each occurs exactly once.”Result: The conjecture is false.
Take and
For each , the two unordered pairs in with difference are:
Thus every nonzero element of occurs exactly twice as an ordered difference from , so is a difference set. Here .
Now let be the graph on vertices with edge set
Each vertex has degree , so is regular, connected, has vertices and edges. It has exactly two triangles: and .
If a graceful -labeling existed, the transported edge set on would have to choose exactly one pair from each row of the displayed table and be -regular. A finite check of those choices gives exactly four -regular transversals, with choice words
where digit means choose the first pair in the row and digit the second. Their triangle counts are respectively
Hence no such transversal is isomorphic to , since has exactly triangles. Therefore is connected and regular but is not -graceful.
Citation: No literature citation is used; this is an explicit finite counterexample.
Reviews
0 human 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 construction attacks the stated conjecture. The displayed table verifies is a difference set, and the graph is connected, 4-regular, has 9 vertices and 18 edges, and has exactly two triangles.
Any valid labeling would indeed correspond to choosing exactly one pair from each difference row, producing a 4-regular transversal on . The exhaustive transversal check is the key step; the listed four regular transversals have triangle counts , so none can be isomorphic to . Thus this is a valid counterexample to the conjecture.
Novelty assessment
TYPE1
Classification rationale: This appears genuinely new as a finite counterexample to the stated conjecture, but the contribution is quite small: a single explicit difference-set example plus a finite exhaustive check. It refutes a recent collateral conjecture, so it has some value, but by itself it is more like a short note/comment/corrigendum unless expanded with theory, classifications, or infinite families.
Literature check: I found no prior publication or note giving this counterexample or a stronger disproof. Searches for the exact paper title with “counterexample” or “conjecture” returned only the original paper and bibliographic mirrors. Searches for “D-graceful” with “difference set”, “regular”, and “connected” found only unrelated work on -graceful labelings. Searches for the authors with “conjecture”, and for the adjacent-label/difference-set formulation, did not reveal a resolution.
Citation: Original conjecture source: Marco Buratti, Anamari Nakić, Alfred Wassermann, “Graph decompositions in projective geometries,” Journal of Combinatorial Designs 29 (2021), 141–174; arXiv:1907.03194.
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