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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.12699 |
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| _version_ | 1866916367382347776 |
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| author | Silberger, Donald |
| author_facet | Silberger, Donald |
| contents | 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 ${\bf x \notin \{b,e\}}$, it happens that in-degree$({\bf x}) =$ out-degree$({\bf x})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12699 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Eulerian Directed Multigraphs Silberger, Donald Combinatorics O5C45 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 ${\bf x \notin \{b,e\}}$, it happens that in-degree$({\bf x}) =$ out-degree$({\bf x})$. |
| title | Eulerian Directed Multigraphs |
| topic | Combinatorics O5C45 |
| url | https://arxiv.org/abs/2408.12699 |