Highly connected orientations from edge-disjoint rigid subgraphs
We give an affirmative answer to a long-standing conjecture of Thomassen, stating that every sufficiently highly connected graph has a k-vertex-connected orientation. We prove that a connectivity of order $O(k^2)$ suffices. As a key tool, we show that for every pair of positive integers d and...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Pi |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050508625000046/type/journal_article |
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| Summary: | We give an affirmative answer to a long-standing conjecture of Thomassen, stating that every sufficiently highly connected graph has a k-vertex-connected orientation. We prove that a connectivity of order
$O(k^2)$
suffices. As a key tool, we show that for every pair of positive integers d and t, every
$(t \cdot h(d))$
-connected graph contains t edge-disjoint d-rigid (in particular, d-connected) spanning subgraphs, where
$h(d) = 10d(d+1)$
. This also implies a positive answer to the conjecture of Kriesell that every sufficiently highly connected graph G contains a spanning tree T such that
$G-E(T)$
is k-connected. |
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| ISSN: | 2050-5086 |