On the number of Euler trails in directed graphs

  • Jakob Jonsson

Abstract

Let $G$ be an Eulerian digraph with all in- and out-degrees equal to 2, and let $\pi$ be an Euler trail in $G$. We consider an intersection matrix $\boldsymbol {L}(\pi)$ with the property that the determinant of $\boldsymbol{L}(\pi) + \boldsymbol{I}$ is equal to the number of Euler trails in $G$; $\boldsymbol{I}$ denotes the identity matrix. We show that if the inverse of $\boldsymbol{L}(\pi)$ exists, then $\boldsymbol{L}^{-1}(\pi) = \boldsymbol{L}(\sigma)$ for a certain Euler trail $\sigma$ in $G$. Furthermore, we use properties of the intersection matrix to prove some results about how to divide the set of Euler trails in a digraph into smaller sets of the same size.
Published
2002-06-01
How to Cite
Jonsson, J. (2002). On the number of Euler trails in directed graphs. MATHEMATICA SCANDINAVICA, 90(2), 191-214. https://doi.org/10.7146/math.scand.a-14370
Section
Articles