On order-compatible paths in infinite graphs

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 $δ$ of edge-disjoint $a{-}b$ paths. G.A. Dirac asked whether there always exists a family of $δ$ 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 $δ$ 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 $δ$ has an affirmative answer if and only if $δ$ has uncountable cofinality. As our second main result, we show that even when Dirac's question fails, it still holds that 'being connected by $δ$ edge-disjoint, pairwise order-compatible paths' is an equivalence relation for all values of $δ$. The most interesting case here is when $δ$ is countable.

Figures (2)

• Figure 1: Choosing the paths $P_{n_0 },P_{n_1},P_{n_2},\ldots$ as in the proof of \ref{['lem:NoTerminals']}.
• Figure 2: Construction of the paths $X_0,X_1,\ldots$, where the orange points correspond to the sequence of cut-vertices $\{w_n\}_{n\in\mathbb{N}}$.

Theorems & Definitions (28)

• Theorem 1.1
• Corollary 1.2
• proof : Proof of \ref{['cor_main']}
• Theorem 1.3
• Lemma 2.1
• proof
• Theorem 2.1
• proof
• Lemma 2.2
• proof