Treewidth order conjecture for token graphs of trees

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Let GG) be a tree on nn vertices, and let Fk(G)F_k(G) denote its kk-token graph, whose vertices are the kk-subsets of V(G)V(G). For a fixed integer kk satisfying 1≤k≤n−11\le k\le n-1, the treewidth order conjecture for token graphs of trees.

tw⁡(Fk(G))∈Θ(nk−1).\operatorname{tw}(F_k(G)) \in \Theta(n^{k-1}).

This extends the established order of growth for token graphs of stars and paths to all trees. The conjecture is presented as open in the source.

References

Primary source

Ruy Fabila-Monroy, Sergio Gerardo Gómez-Galicia, César Hernández-Cruz and Ana Laura Trujillo-Negrete, “On the Treewidth of Token and Johnson Graphs”, arXiv:2402.17962 (2025).

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