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On the minimum number of spanning trees in \(k\)-edge-connected graphs. (English) Zbl 1359.05023
Summary: We show that a \(k\)-edge-connected graph on \(n\) vertices has at least \(n(k/2)^{n-1}\) spanning trees. This bound is tight if \(k\) is even and the extremal graph is the \(n\)-cycle with edge multiplicities \(k/2\). For \(k\) odd, however, there is a lower bound \(c_k^{n-1}\), where \(c_k>k/2\). Specifically, \(c_3>1.77\) and \(c_5>2.75\). Not surprisingly, \(c_3\) is smaller than the corresponding number for 4-edge-connected graphs. Examples show that \(c_3\leq\sqrt{2+\sqrt{3}}\approx1.93\). However, we have no examples of 5-edge-connected graphs with fewer spanning trees than the \(n\)-cycle with all edge multiplicities (except one) equal to 3, which is almost 6-regular. We have no examples of 5-regular 5-edge-connected graphs with fewer than \(3.09^{n-1}\) spanning trees, which is more than the corresponding number for 6-regular 6-edge-connected graphs. The analogous surprising phenomenon occurs for each higher odd edge connectivity and regularity.

MSC:
05C05 Trees
05C40 Connectivity
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