On the Minimum Number of Spanning Trees in k-Edge-Connected Graphs

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 cn−1 k where ck > k/2. Specifically, c3 > 1.77 and c5 > 2.75. Not surprisingly, c3 is smaller than the corresponding number for 4-edge-connected graphs. Examples show that c3 ≤ √ 2 + √ 3 ≈ 1.93. 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 3.09n−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.