Szegö's construction conjecture for tight multigraphs

From papers

Let Γ=(V,E)\Gamma=(V,E) be a (d,)(d,\ell)-tight multigraph, with <4d+23\ell<\frac{4d+2}{3}. Let K22kK_2^{2k-\ell} denote the graph with two vertices and 2k2k-\ell parallel edges, and let a dd-dimensional kk-extension be the graph operation used in the conjecture. Szegö's conjecture. The graph Γ\Gamma can be constructed from K22kK_2^{2k-\ell} using dd-dimensional kk-extensions, with no more than 2d2d-\ell parallel edges created between any pair of vertices. This conjecture strengthens the Frank–Szegö construction theorem by imposing the stated bound on parallel edges for (d,)(d,\ell)-tight multigraphs in the indicated range of \ell; its resolution is not specified in the source.

Progress summary

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Sources & referencesView supporting material

Primary source

Joannes Vermant, “Homological methods in rigidity theory using graphs of groups”, arXiv:2603.05435 (2026).

Solutions 0

No solutions have been posted yet.