Towards Nash-Williams Orientation Conjecture for Infinite Graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914926003486720 |
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| author | Assem, Amena |
| author_facet | Assem, Amena |
| contents | In 1960 Nash-Williams proved that an edge-connectivity of 2k is sufficient for a finite graph to have a k-arc-connected orientation. He then conjectured that the same is true for infinite graphs. In 2016, Thomassen, using his own results on the auxiliary lifting graph, proved that 8k-edge-connected infinite graphs admit a $k$-arc connected orientation. Here we improve this result for the class of $1$-ended locally-finite graphs and show that an edge-connectivity of 4k is enough in that case. Crucial to this improvement are results presented in a separate paper, by the same author of this paper, on the key concept of the lifting graph, extending results by Ok, Richter, and Thomassen. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_10631 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Towards Nash-Williams Orientation Conjecture for Infinite Graphs Assem, Amena Combinatorics In 1960 Nash-Williams proved that an edge-connectivity of 2k is sufficient for a finite graph to have a k-arc-connected orientation. He then conjectured that the same is true for infinite graphs. In 2016, Thomassen, using his own results on the auxiliary lifting graph, proved that 8k-edge-connected infinite graphs admit a $k$-arc connected orientation. Here we improve this result for the class of $1$-ended locally-finite graphs and show that an edge-connectivity of 4k is enough in that case. Crucial to this improvement are results presented in a separate paper, by the same author of this paper, on the key concept of the lifting graph, extending results by Ok, Richter, and Thomassen. |
| title | Towards Nash-Williams Orientation Conjecture for Infinite Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2306.10631 |