In our earlier paper [9], generalizing the well known notion of graceful graphs, a (p, m, n)-signed graph S of order p, with m positive edges and n negative edges, is called graceful if there exists an injective function f that assigns to its p vertices integers 0, 1,…,q = m + n such that when to each edge uv of S one assigns the absolute difference |f(u)-f(v)| the set of integers received by the positive edges of S is {1,2,…,m} and the set of integers received by the negative edges of S is {1,2,…,n}. Considering the conjecture therein that all signed cycles Z_k, of admissible length k ≥ 3 and signed structures, are graceful, we establish in this paper its truth for all possible signed cycles of lengths 0, 2 or 3 (mod 4) in which the set of negative edges forms a connected subsigraph.
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机译：在我们之前的论文[9]中，概括了优美图的概念，如果存在一个内射词，则具有m个正边和n个负边的阶p的（p，m，n）有符号图S被称为优美。函数f，向其p个顶点分配整数0、1，…，q = m + n，以便当向S的每个边uv分配绝对差| f（u）-f（v）|时， S的上升沿接收的整数集为{1,2，…，n}，S的下降沿接收的整数集为{1,2，…，n}。考虑到其中的猜想，所有长度为k≥3的有符号周期Z_k和有符号结构都是优美的，因此我们在本文中确定了长度为0、2或3（模4）的所有可能有符号周期的真相，其中负边缘的个数形成一个相连的子图。
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