On order-compatible paths in infinite graphs

TL;DR

This paper proves that in infinite graphs, a large set of edge-disjoint paths can be ordered compatibly under certain conditions, confirming a conjecture and exploring the relation between path orderability and infinite cardinalities.

Contribution

It confirms Zelinka's conjecture for bounded-length paths and characterizes when Dirac's question has an affirmative answer based on the cofinality of the cardinality.

Findings

Confirmed Zelinka's conjecture for bounded-length paths

Characterized when Dirac's question is affirmative based on cofinality

Established that order-compatibility forms an equivalence relation for all cardinalities

Abstract

Two $a{-}b$ paths in a graph $G$ are order-compatible if their common vertices occur in the same order when travelling from $a$ to $b$. Suppose a graph contains an infinite number $\delta$ of edge-disjoint $a{-}b$ paths. G.A. Dirac asked whether there always exists a family of $\delta$ edge-disjoint $a{-}b$ paths that are pairwise order-compatible. Confirming a conjecture by B. Zelinka, we show that this holds provided that the given $\delta$ edge-disjoint $a{-}b$ paths have bounded length. Combining this with an earlier work of Zelinka, it follows that Dirac's question for an infinite cardinal $\delta$ has an affirmative answer if and only if $\delta$ has uncountable cofinality. As our second main result, we show that even when Dirac's question fails, it still holds that 'being connected by $\delta$ edge-disjoint, pairwise order-compatible paths' is an equivalence relation for all values of $\delta$. The most interesting case here is when $\delta$ is countable.