发表机构
School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在聚合度条件下证明了图系统的Bondy型定理：若图系统含彩虹Hamilton圈且最小度之和达到阈值，则系统为彩虹泛圈，除非偶数情形下的平衡完全二部图例外。
AI 中文摘要
我们在聚合度条件下建立了图系统中Bondy定理的Hamilton到泛圈的类比。设 $\G=(G_1,\ldots,G_n)$ 是共同 $n$ 顶点集 $V$ 上的图系统，并记 $δ(v)=\min_{i\in[n]}d_{G_i}(v)$。若 $\G$ 包含一个彩虹Hamilton圈且 \\[ \sum_{v\in V}δ(v)\ge \left\lceil\frac{n^2}{2}\right\rceil-1, \\] 则 $\G$ 是彩虹泛圈的，除非 $n$ 为偶数且每个成员都是相同的平衡完全二部图。对于偶数 $n$，该阈值在整数层面是精确的。与通常的横截Dirac型或Ore型假设不同，我们的条件不是逐层的：达到 $δ(v)$ 的成员可能依赖于 $v$，且某些顶点可能有 $δ(v)<n/2$。相对于固定的彩虹Hamilton圈，我们计算其颜色被所替换的Hamilton弧释放的捷径。缺失的圈长迫使互补的捷径支撑集交叉相交。一个计数间隙解决了偶数缩短，而奇数缩短中的相等或接近相等产生距离二的交换，其轨道迫使平衡二部障碍。在较低的整数阈值处，一个精确的缺陷恒等式表明只有一或两个单位的松弛可用。\noindent\textbf{关键词：} 图系统；彩虹圈；泛圈性；Hamilton圈；极值图论。
英文摘要
We establish a Hamiltonian-to-pancyclic analogue of Bondy's theorem for graph systems under an aggregate degree condition. Let $\G=(G_1,\ldots,G_n)$ be a graph system on a common $n$-vertex set $V$, and write $δ(v)=\min_{i\in[n]}d_{G_i}(v)$. If $\G$ contains a rainbow Hamilton cycle and \[ \sum_{v\in V}δ(v)\ge \left\lceil\frac{n^2}{2}\right\rceil-1, \] then $\G$ is rainbow pancyclic, unless $n$ is even and every member is the same balanced complete bipartite graph. For even $n$ the threshold is exact at the integer level. Unlike the usual transversal Dirac- or Ore-type hypotheses, our condition is not layerwise: the member attaining $δ(v)$ may depend on $v$, and some vertices may have $δ(v)<n/2$. Relative to a fixed rainbow Hamilton cycle, we count shortcuts whose colors are released by the Hamilton arcs they replace. A missing cycle length forces complementary shortcut supports to cross-intersect. A counting gap settles even shortening, while equality or near equality in odd shortening yields a distance-two exchange whose orbits force the balanced bipartite obstruction. At the lower integer threshold an exact defect identity shows that only one or two units of slack are available. \noindent\textbf{Keywords:} graph system; rainbow cycle; pancyclicity; Hamilton cycle; extremal graph theory.
