On orientations preserving edge-connectivity in infinite graphs

TL;DR

The paper extends classical edge-connectivity orientation results to graphs with countably many edge-ends by developing bond-faithful subgraph techniques and local-to-global reductions. It proves that every $2k$-edge-connected graph with countably many edge-ends admits a $k$-arc-connected orientation, via a locally finite expansion and a decomposition scheme that lifts orientations from countable subgraphs. It also establishes a strong-topological variant using ETOP spaces, and proves that the strong conjecture can be reduced to the locally finite case, paralleling the weak conjecture. Overall, the work introduces a unified framework combining bond-faithful reductions, expanding-ray constructions, and topological methods to address infinite-graph orientation problems with edge-end considerations.

Abstract

We prove that every 2k-edge-connected graph with countably many edge-ends admits a k-arc-connected orientation, extending the previous result by Assem, Koloschin and Pitz that also assumed the hypothesis of the graph being locally finite. We prove that, if every locally finite graph has a well-balanced orientation, so does every graph. Lastly, we explore an alternative to the Nash-Williams Orientation Conjecture via topological paths, and prove that it is true for every finitely separated graph.

Figures (9)

• Figure 1: The graph $G$
• Figure 2: Representing a $7$-ray
• Figure 3: The Expansion of an infinite vertex into a $4$-ray
• Figure 4: Transforming a path on $\widetilde{G}$ into a trail of $G$.
• Figure 5: Representing an expanding-ray.

Theorems & Definitions (61)

• Conjecture 1.1: Strong Conjecture
• Conjecture 1.2: Weak Conjecture
• Theorem : \ref{['thm:general']}
• Theorem : \ref{['thm:wellb']}
• Definition 2.1
• Example 2.2
• proof
• Definition 2.3
• Lemma 2.4
• proof