The minimum number of spanning trees in regular multigraphs I: the odd-degree case

Nombre minimum d'arbres couvrants des multigraphes réguliers I : degré impair

Jakub Pekárek (1) , Jean-Sébastien Sereni (2) , Zelealem B Yilma (3)

1 IUUK - Computer Science Institute of Charles University [Prague]
2 CNRS - Centre National de la Recherche Scientifique
3 Carnegie Mellon University in Qatar

In a recent article, Bogdanowicz determines the minimum number of spanning trees a connected cubic multigraph on a fixed number of vertices can have and identifies the unique graph that attains this minimum value. He conjectures that a generalized form of this construction, which we here call a padded paddle graph, would be extremal for d-regular multigraphs where d ≥ 5 is odd. We prove that, indeed, the padded paddle minimises the number of spanning trees, but this is true only when the number of vertices, n, is greater than (9d+6)/8. We show that a different graph, which we here call the padded cycle, is optimal for n < (9d+6)/8. This fully determines the d-regular multi-graphs minimising the number of spanning trees for odd values of d. The approach we develop can also be applied to the even-degree case. However, the extremal structures are more irregular, and the slightly more technical analysis is done in a companion article.

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Jakub Pekárek, Jean-Sébastien Sereni, Zelealem B Yilma. The minimum number of spanning trees in regular multigraphs I: the odd-degree case. 2021. ⟨hal-03472264⟩