The diameter of almost Eulerian digraphs
Summary: \textit{J. Soares} [J. Graph Theory 16, No.~5, 437--450 (1992; Zbl 0768.05048)] showed that the well known upper bound \(\frac{3}{\delta+1}n+O(1)\) on the diameter of undirected graphs of order \(n\) and minimum degree 5 also holds for digraphs, provided they are Eulerian. In this paper we investigate if similar bounds can be given for digraphs that are, in some sense, close to being Eulerian. In particular we show that a directed graph of order \(n\) and minimum degree \(\delta\) whose arc set can be partitioned into s trails, where \(s\leq\delta-2\), has diameter at most \(3(\delta+1-\frac s3)^{-1}n+ O(1)\). If s also divides \(\delta-2\), then we show the diameter to be at most \(3(\delta+1- \frac{(\delta-2)s}{3(\delta-2)+s})^{-1}n+ O(1)\). The latter bound is sharp, apart from an additive constant. As a corollary we obtain the sharp upper bound \(3(\delta+1- \frac{\delta-2}{3\delta-5})^{-1}n+ O(1)\) on the diameter of digraphs that have an Eulerian trail.
scientific article; zbMATH DE number 5827406