发表机构
SuZhou University of Science and Technology; Tianjin University（苏州科技大学; 天津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文刻画连通符号图的欧拉亏格谱，证明每个奇偶类内亏格构成步长二区间且两类的最大值相差一，从而肯定回答Širáň关于谱间隙位置的提问，方法涉及预符号图表示、有序相邻交换及面数匹配解释。
AI 中文摘要
Duke插值定理指出，连通图的可定向亏格构成一个整数区间，Stahl建立了非可定向嵌入的相应结果。1991年，Širáň证明了该插值性质对符号图嵌入不成立：符号图的欧拉亏格谱可能含有间隙。他随后提出疑问：所有这些间隙是否必然出现在谱的下端。本文建立了连通符号图的欧拉亏格谱的刻画。我们证明，对每个奇偶类，欧拉亏格构成步长为二的区间。此外，当两个奇偶类均非空时，它们的最大元素相差一。作为推论，若两个连续整数k和k+1属于欧拉亏格谱，则从k到最大欧拉亏格的所有整数也属于该谱，从而肯定地回答了Širáň的问题。我们的证明使用了符号嵌入的预符号图表示，结合有序相邻交换操作和面数的匹配解释。
英文摘要
Duke's interpolation theorem states that the orientable genera of a connected graph form an integer interval, and Stahl established the corresponding result for nonorientable embeddings. In 1991, Širáň showed that this interpolation property fails for signed graph embeddings: the Euler-genus spectrum of a signed graph may contain gaps. He subsequently asked whether all such gaps must occur at the lower end of the spectrum. In this paper, we establish a characterization of the Euler-genus spectrum of a connected signed graph. We prove that, for each parity class, the Euler genera form a step-two interval. Moreover, whenever both parity classes are nonempty, their maximum elements differ by one. As an consequence, if two consecutive integers $k$ and $k+1$ belong to the Euler-genus spectrum, then every integer from $k$ to the maximum Euler-genus also belongs to the spectrum, thereby answering Širáň's question affirmatively. Our proof uses the pre-signed graph representation of signed embeddings together with ordered adjacent-exchange operations and a matching interpretation of face numbers.
Comments24 pages, 2 figures
