Distinguishing regular graphs from lists

TL;DR

This work addresses distinguishing edge colourings from lists in connected regular graphs, proving that for graphs of order at least $7$ (finite or countable) there exists a distinguishing edge colouring from lists of length $2$, i.e. $D'_l(G) ≤ 2$, with an analogous extension for $\kappa$-regular graphs where $\kappa$ is a fixed point of the aleph hierarchy. It develops a constructive approach: a rooted-automorphism framework with a phase-based colouring for locally finite graphs, and a vertex-palette encoding method for graphs of infinite degree, to guarantee a two-colour distinguishing colouring from lists. The paper also establishes $D'_l(K_n)=2$ for $n≥6$ via asymmetric spanning subgraphs and discusses forcing-based consistency results for $2^{\kappa}$-regular graphs, broadening the applicability to both finite and infinite regular graphs. These results advance the understanding of list-distinguishing indices, connect to the Infinite Motion Conjecture, and provide a versatile framework for automorphism-breaking in regular graphs across cardinalities.

Abstract

An edge colouring of a graph is called distinguishing if there is no non-trivial automorphism which preserves it. We prove that every at most countable, finite or infinite, connected regular graph of order at least $7$ admits a distinguishing edge colouring from any set of lists of length $2$. Furthermore, we show that the same holds for connected regular graphs of order $κ$ where $κ$ is a fixed point of the aleph hierarchy.

Theorems & Definitions (21)

• Theorem 1: Kwaśny, Stawiski KS_list_general
• Conjecture 2
• Theorem 3
• Theorem 4
• Theorem 5
• Conjecture 6
• Conjecture 7: Broere, Pilśniak 2015 BP
• Conjecture 8
• Lemma 9
• proof