图中$S^1$-流的归约操作与结构刻画
Reduction Operations and Characterizations of $S^1$-Flows in Graphs
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中文总结 AI 辅助
本文针对图中$S^1$-流开发归约技术，利用双端$S^1$-预流证明相关结论，并完全刻画三角连通图、含生成三角树的图的$S^1$-流存在条件。
中文摘要 AI 辅助
尽管每个容许非零3-流的图也容许$S^1$-流，但Thomassen（2014）证明，反之并不普遍成立。本文基于牛图增长、2-和、轮收缩等图操作，开发了$S^1$-流的归约技术。关键工具是双端$S^1$-预流，它使我们能证明：若2-连通图包含奇轮作为真子图，且收缩该轮后得到的图有非零3-流，则原图容许$S^1$-流。作为应用，我们完全刻画了两类图的$S^1$-流：三角连通图容许$S^1$-流当且仅当它不是奇轮；含生成三角树的图容许$S^1$-流当且仅当它不是奇晶体。
英文摘要
Thomassen (J. Combin. Theory Ser. B 108 (2014), 81-91) showed that every graph admitting a nowhere-zero $3$-flow also admits an $S^1$-flow. He also proved that the converse holds for cubic graphs, but constructed counterexamples showing it fails in general. Wang et al. (SIAM J. Discrete Math. 29 (2015), 2166-2178) presented a couple of sufficient conditions under which the existence of an $S^1$-flow guarantees the existence of a nowhere-zero 3-flow. In this paper, we first prove that a graph with maximum degree at most four admits a nowhere-zero $3$-flow if and only if it admits an $S^1$-flow. We then develop some reduction techniques for $S^1$-flows based on graph operations including bull-growth, $2$-sums, and contractions. Finally, we apply those techniques to characterize triangularly connected graphs and graphs containing a spanning triangle-tree that admit $S^1$-flows, respectively.
发表机构
- School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院）
- Department of Mathematics, West Virginia University（西弗吉尼亚大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
