Abstract A signed edge domination function (or SEDF) of a simple graph G = ( V , E ) is a function f : E → { 1 , − 1 } such that ∑ e ′ ∈ N [ e ] f ( e ′ ) ≥ 1 holds for each edge e ∈ E , where N [ e ] is the set of edges in G that share at least one endpoint with e . Let γ s ′ ( G ) denote the minimum value of f ( G ) among all SEDFs f , where f ( G ) = ∑ e ∈ E f ( e ) . In 2005, Xu conjectured that γ s ′ ( G ) ≤ n − 1 , where n is the order of G . This conjecture has been proved for the two cases v o d d ( G ) = 0 and v e v e n ( G ) = 0 , where v o d d ( G ) (resp. v e v e n ( G ) ) is the number of odd (resp. even) vertices in G . This article proves Xu’s conjecture for v e v e n ( G ) ∈ { 1 , 2 } . We also show that for any simple graph G of order n , γ s ′ ( G ) ≤ n + v o d d ( G ) ∕ 2 and γ s ′ ( G ) ≤ n − 2 + v e v e n ( G ) when v e v e n ( G ) > 0 , and thus γ s ′ ( G ) ≤ ( 4 n − 2 ) ∕ 3 . Our result improves the best current upper bound of γ s ′ ( G ) ≤ ⌈ 3 n ∕ 2 ⌉ .

中文翻译：

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图的有符号边控制数的上限

摘要 简单图 G = ( V , E ) 的有符号边支配函数 (或 SEDF) 是函数 f : E → { 1 , − 1 } 使得 ∑ e ′ ∈ N [ e ] f ( e ′ ) ≥ 1 对于每条边 e ∈ E 成立，其中 N [ e ] 是 G 中与 e 共享至少一个端点的边的集合。令γ s ′ ( G ) 表示所有SEDF f 中f ( G ) 的最小值，其中f ( G ) = ∑ e ∈ E f ( e ) 。2005 年，徐推测γ s ′ ( G ) ≤ n − 1 ，其中n 是G 的阶数。这个猜想已经在 vodd ( G ) = 0 和 veven ( G ) = 0 两种情况下得到证明，其中 vodd ( G ) (resp. veven ( G ) ) 是 G 中奇数 (resp. even) 顶点的数量。本文证明了徐对 veven ( G ) ∈ { 1 , 2 } 的猜想。我们还表明，对于任何 n 阶简单图 G，γ s ′ ( G ) ≤ n + vodd ( G ) ∕ 2 且 γ s ′ ( G ) ≤ n − 2 + veven ( G ) 当 veven ( G ) > 0 时，因此 γ s ′ ( G ) ≤ ( 4 n − 2 ) ∕ 3 。我们的结果改进了 γ s ′ ( G ) ≤ ⌈ 3 n ∕ 2 ⌉ 的当前最佳上限。