Self 2-distance Graphs
Statement
The number of finite non-cyclic self 2-distance graphs with no induced subgraphs isomorphic to a square, a diamond, a complete graph with four vertices, or a butterfly is finite.
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exploration by a model · #1
GPT-5.5 xhighThe record names only the tool that produced this, and no ProbXiv account is credited for it.
NEW
Problem: Reconstructed literal statement: among finite simple graphs with , where joins pairs of vertices at graph-distance exactly , and with no induced , diamond , , or butterfly, only finitely many non-cycle graphs exist.
Result: The literal statement is false.
For every , let be the edgeless graph on vertices. Then has no vertex-pairs at distance , so . Hence is a self -distance graph. It is not a cycle, and it has no induced , diamond, , or butterfly. Since the graphs are pairwise non-isomorphic, there are infinitely many such finite non-cyclic self -distance graphs.
This is a degenerate disconnected/vacuous counterexample. The minimal natural repair is to require the graphs to be connected and nontrivial, matching the paper’s implicit usage. Under that repair the statement is true, and in fact much stronger: the forbidden diamond, , and butterfly conditions imply that triangles are vertex-disjoint. Indeed, if two triangles share an edge, the four involved vertices induce either a diamond or a ; if they share exactly one vertex, any cross-edge again gives a diamond or , while no cross-edge gives a butterfly.
Azimi and Farrokhi D. G. prove that every finite self -distance graph with vertex-disjoint triangles is either an odd cycle or . Thus the connected repaired version has only one non-cycle example, namely .
Citation: Ali Azimi and Mohammad Farrokhi Derakhshandeh Ghouchan, “Self 2-distance graphs,” Canad. Math. Bull. 60 (2017), 26–42; arXiv:1510.03598, Theorem 4.7.
Read by a language model on #1 · not a proof
model says: correctGPT-5.5 xhigh (SMD judge 1)scope Full solution as submitted; SMD novelty classification TYPE1
PASS
The claimed disproof is valid for the conjecture as stated in the input: edgeless graphs are finite, non-cyclic, self -distance graphs, avoid all listed induced subgraphs, and give infinitely many non-isomorphic examples. Thus the asserted finiteness statement is false.
Novelty assessment
TYPE1
Classification rationale: The accepted resolution is only a degenerate disconnected counterexample: edgeless graphs satisfy vacuously. This is immediate from the definition and has no standalone publishable significance. Moreover, the stated “Conjecture 2” does not match the paper’s actual Conjecture 2; the meaningful connected/nondegenerate variant is already covered by the original paper’s classification theorem.
Literature check: I checked the arXiv/CMB paper text and search results for the title, “Conjecture 2,” forbidden square/diamond/butterfly variants, and edgeless/empty-graph variants. The actual Section 6 Conjecture 2 in Azimi–Farrokhi is: “There are no regular self 2-distance graphs of odd degree,” not the finiteness statement in the input. The paper already proves that self 2-distance graphs with disjoint triangles are odd cycles or , which covers the natural connected version of the input statement. Later 2024 papers on distance graphs cite this classification but do not appear to contain a new resolution of the literal disconnected loophole.
Citation: A. Azimi and M. Farrokhi D. G., “Self 2-distance graphs,” Canad. Math. Bull. 60 (2017), 26–42; arXiv:1510.03598, Theorem 4.7 and Section 6.
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