二分有向图中的反馈边集

中文摘要 AI 辅助

设 \\(\beta(G)\\) 表示有向图 \\(G\\) 的反馈边集的最小大小，设 \\(\gamma(G)\\) 表示非相邻顶点的无序对的数量。受 Chudnovsky--Seymour--Sullivan 关于 \\(3\\)-自由有向图的猜想的启发，我们研究了二分有向图的相应反馈边问题。在二分情形中，\\(\gamma(G)\\) 仅计算端点位于不同部集中的非相邻对。我们证明了每个 \\(4\\)-自由二分有向图 \\(G\\) 满足 \\(\beta(G)\le \gamma(G)/2\\)。我们还确定了具有部集 \\(X\\) 和 \\(Y\\) 的 \\(2k\\)-自由强二分有向图的精确 Turán 数：若 \\(|X|,|Y|\ge k+1\\)，则最大边数为 $$(|X|-(k-1))(|Y|-(k-1))+2k-2.$$ 最后，对于极值情形 \\(k=2\\)，我们分析了 \\(4\\)-自由强二分 Turán 有向图的结构，并证明了所有此类有向图满足更精确的界 \\(\beta(G)\le \gamma(G)/3\\)。该常数由自然的三块平衡构造达到。

英文摘要

Let \(β(G)\) denote the minimum size of a feedback edge set of a digraph \(G\), and let \(γ(G)\) denote the number of unordered pairs of nonadjacent vertices. Motivated by the Chudnovsky--Seymour--Sullivan conjecture for \(3\)-free digraphs, we study the corresponding feedback-edge problem for bipartite digraphs. In the bipartite setting, \(γ(G)\) is taken to count only nonadjacent pairs with ends in distinct partite sets. We prove that every \(4\)-free bipartite digraph \(G\) satisfies \(β(G)\le γ(G)/2\). We also determine the exact Turán number of \(2k\)-free strong bipartite digraphs with partite sets \(X\) and \(Y\): if \(|X|,|Y|\ge k+1\), then the maximum number of edges is $$(|X|-(k-1))(|Y|-(k-1))+2k-2.$$ Finally, for the extremal case \(k=2\), we analyze the structure of \(4\)-free strong bipartite Turán digraphs and prove the sharper bound \(β(G)\le γ(G)/3\) for all such digraphs. This constant is attained by a natural balanced three-block construction.

发表机构

• School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）
• Center for Discrete Mathematics, Fuzhou University（福州大学离散数学中心）

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