2024/08/22 by Donald Silberger, Silberger, Donald
Computer Science · Engineering · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #O5C45 #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2408.12699
openalex publication_date 2024/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For Δ a finite connected nontrivial directed multigraph, we prove: 1. Δ has a directed circuit using each directed edge exactly once if and only if both each pair of distinct vertices of Δ occur in a common directed circuit and in-degree(\bf x) = out-degree(\bf x) for every vertex \bf x. 2. Δ contains a non-circuit directed path which uses every directed edge exactly once if and only if both every pair of distinct vertices of Δ occur in a common directed circuit and there are vertices \bf b \not= e such that in-degree(\bf e) - out-degree(\bf e) = 1 = out-degree(\bf b) - in-degree(\bf b) but, for every vertex \bf x ∉ \b,e\, it happens that in-degree(\bf x) = out-degree(\bf x).