Huang, Shiying; Liu, Yuelin Tree number pairs for free and wired spanning forests. (English) Zbl 1488.05057 Acta Sci. Nat. Univ. Nankaiensis 54, No. 2, 27-30 (2021). Summary: Let \(N\) be the set of pairs \((k, l) \in ({\mathrm{N}} \cup \{\infty\})^2\) such that on some infinite electric network, wired spanning forest has \(l\) trees almost surely, while free spanning forest has \(k\) trees with positive probability. It’s proved that \(N = \{(k, l) \in {{\mathrm{N}}^2}\mid k \le l\} \cup \{(\infty, \infty)\} \cup \{(k, \infty)\mid k \in {\mathrm{N}}\}\). MSC: 05C05 Trees Keywords:free spanning forest; wired spanning forest; tree number pair PDFBibTeX XMLCite \textit{S. Huang} and \textit{Y. Liu}, Acta Sci. Nat. Univ. Nankaiensis 54, No. 2, 27--30 (2021; Zbl 1488.05057)