Spanning Trees of Bounded Degree Graphs

John Michael Robson (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

We consider lower bounds on the number of spanning trees of connected graphs with degree bounded by $d$. The question is of interest because such bounds may improve the analysis of the improvement produced by memorisation in the runtime of exponential algorithms. The value of interest is the constant $\beta_d$ such that all connected graphs with degree bounded by $d$ have at least $\beta_d^\mu$ spanning trees where $\mu$ is the cyclomatic number or excess of the graph, namely $m-n+1$. We conjecture that $\beta_d$ is achieved by the complete graph $K_{d+1}$ but we have not proved this for any $d$ greater than $3$. We give weaker lower bounds on $\beta_d$ for $d\le 11$.

Mots clés: spanning trees; memorisation; cyclomatic number; bounded degree graphs; cut; linear program

Domaines: Mathématique discrète [cs.DM]; Complexité [cs.CC]

John Michael Robson. Spanning Trees of Bounded Degree Graphs. 2008. ⟨hal-00360110⟩