A generalization of a theorem of Diderrich in additive group theory to vertex-transitive graphs
Consider a vertex-transitive (finite) directed graph \(X=(V,E)\). Let \(\kappa (X)\) be its connectivity number in directed sense. As known, each indegree and each outdegree in \(X\) equals \(|E|/ |X|\). Denote this number by \(d\). It is shown that \(d= \kappa (X)\) if there is no transitive triangle in \(X\). Let a sequence \(X_i= (V,E_i)\) of directed graphs be given where \(i\) runs from 1 to \(k\). Assume that each of these graphs is vertex-transitive and contains no transitive triangle. For \(a\in V\) denote by \(\Lambda (a)\) the set of all vertices \(b\) satisfying the following requirement: there exist a number \(t(\leq k)\), an increasing sequence \((1\leq) i_1< i_2<\cdots <i_t (\leq k)\), and a path \(P\) of length \(t\) such that \(P\) leads from \(a\) to \(b\), and, for any \(j\), the \(j\)th edge of \(P\) belongs to \(E_{ij}\). It is proved that then either \(\Lambda (a)\) is included in the vertex set of a connected component of some \(X_i\) or \(|\Lambda (a) |> d_1+ d_2+ \cdots + d_k\) where \(d_i\) is the common semidegree in \(X_i\). A corollary about finite groups is deduced from this graph-theoretical theorem. The corollary is a refinement of Theorem 3 of \textit{G. T. Diderrich} in [Proc. Am. Math. Soc. 38, 443-451 (1973; Zbl 0266.20041)].
scientific article; zbMATH DE number 944214