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All $\tau_k$ -maximal graphs have the same size

Contributed by Deep Bhattacharjee

A graph is $\tau_k$ -maximal if none of its subgraphs has $k+1$ edge-disjoint spanning trees, but adding any missing edge creates such a subgraph. Wang and Tian conjectured that every $\tau_k$ -maximal graph on $n\ge 2k+2$ vertices has exactly $(k+1)(n-1)-1$ edges. We prove this conjecture.

A mix of human-written and AI-generated textHuman understanding: some partscount matroidedge-disjoint spanning treessaturated graphssparse graphs

Oct 22, 2026PDF ↗View work →

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All $\tau_k$ -maximal graphs have the same size

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All τk\tau_kτk​-maximal graphs have the same size

All $\tau_k$ -maximal graphs have the same size