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arXiv · 2504.00940

Edge-disjoint linkage in infinite graphs

Amena Assem·R. Bruce Richter
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Abstract

In 1980, Thomassen stated his weak linkage conjecture: for an odd positive integer k, if a graph G is k-edge-connected, then, for any collection of k pairs of vertices {s_1,t_1}, ..., {s_k,t_k} in G, not necessarily distinct, there are pairwise edge-disjoint paths P_1,...,P_k in G, with P_i joining s_i and t_i. In 1991, Huck proved that the conclusion holds if G is finite and (k+1)-edge-connected. We prove that Huck's theorem holds also for all infinite graphs, extending and improving a result of Ok, Richter and Thomassen for 1-ended, locally finite graphs. A novel key tool in the proof is the Linking Fan Proposition proved in Section 3. To show the potential and usefulness of this proposition in other contexts, we apply it in the last section to prove a new result, similar to a result of Thomassen, on the existence of 2k-edge-connected finite immersions in (2k+1)-edge-connected infinite graphs. We then use this to prove that an edge-connectivity of 2k+1 is sufficient for infinite graphs to admit a k-arc-connected orientation. This is only within 1 of the longstanding conjecture of Nash-Williams from 1960 that an edge-connectivity of 2k should be enough.

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Amena Assem, R. Bruce Richter. 2025-04-01. Edge-disjoint linkage in infinite graphs. https://arxiv.org/abs/2504.00940

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