Edge-disjoint linkage in infinite graphs

TL;DR

The paper extends Huck's finite-case result that $(k+1)$-edge-connected graphs are weakly $k$-linked to all infinite graphs for odd $k$, by developing a refined lifting technique for locally finite graphs and proving a new Linking Fan Proposition. This enables reducing the infinite problem to finite instances and then repairing the linkage in the original graph via boundary-linked decompositions and compatible lifts. A key auxiliary contribution is a new immersion theorem and its use to derive highly connected immersions and, ultimately, a $k$-arc-connected orientation in infinite graphs under $4k$-edge-connectivity. The results broaden linkage theory and orientation theory in the infinite setting, offering tools that may apply to other edge-connectivity questions in infinite graphs.

Abstract

In 1980, Thomassen stated his Weak Linkage conjecture: for odd positive integers $k$, 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 present in the last section a new result, similar to a result of Thomassen, on the existence of $2k$-edge-connected finite immersions in $(2k+2)$-edge-connected infinite graphs, and as a corollary we also reprove a statement on $k$-arc-connected orientations.

Theorems & Definitions (18)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof