Abstract MacDougall’s conjecture states that every regular graph of degree at least 2 has a vertex-magic total labeling (VMTL) with the lone exception of 2 K 3 . Since there is enormous empirical evidence supporting this conjecture, it is reasonable to seek generalizations. Thus we ask the more general question: to what extent does the degree sequence of a graph determine the existence or nonexistence of a VMTL? We provide beginning steps towards answering this question, and related questions, by providing infinite families of degree sequences, and for each sequence, a graph with a VMTL and another graph without a VMTL.

中文翻译：

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对 MacDougall 对顶点魔法总标记的猜想进行概括

摘要 MacDougall 猜想指出，每个度数至少为 2 的正则图都有一个顶点魔术全标记 (VMTL)，只有 2 K 3 例外。由于有大量经验证据支持这一猜想，因此寻求概括是合理的。因此，我们提出了一个更普遍的问题：图的度数序列在多大程度上决定了 VMTL 的存在或不存在？我们提供了回答这个问题和相关问题的开始步骤，方法是提供无限的度序列族，对于每个序列，一个带有 VMTL 的图和另一个没有 VMTL 的图。