二部广义超立方体的二不相交圈覆盖边双泛圈性

中文摘要 AI 辅助

设 \\(G=C(d_1,\ldots,d_n)=F_1\BoxProd\cdots\BoxProd F_n\\) 为一个二部广义超立方体，其中 \\(n\geq2\\)，所有 \\(d_i\\) 为偶数，且 \\(N=|V(G)|\geq8\\)，这里当 \\(d_i=2\\) 时 \\(F_i=K_2\\)，当 \\(d_i\geq4\\) 时 \\(F_i=C_{d_i}\\)。我们证明了二不相交圈覆盖顶点双泛圈性的如下精确加强。对于每一对独立边 \\(e,f\in E(G)\\) 和每个偶数 \\(4\leq\ell\leq N-4\\)，顶点集可以划分为两个圈 \\(J_1,J_2\\)，其长度分别为 \\(\ell\\) 和 \\(N-\ell\\)，且满足 \\(e\in E(J_1)\\) 和 \\(f\in E(J_2)\\)，当且仅当 \\(G\\) 不同构于任何 \\(K_2\BoxProd C_{2p}\\)（其中 \\(p\geq3\\)）。图 \\(C(2,2)\cong C_4\\) 单独处理：它没有 2-DCC。保留指定边信息的推论包括普通边双泛圈性和偶数 \\(k\\) 元 \\(n\\) 立方体的特例。

英文摘要

Let \(G=C(d_1,\ldots,d_n)=F_1\BoxProd\cdots\BoxProd F_n\) be a bipartite generalized hypercube with \(n\geq2\), all \(d_i\) even, and \(N=|V(G)|\geq8\), where \(F_i=K_2\) when \(d_i=2\), and \(F_i=C_{d_i}\) when \(d_i\geq4\). We prove the following exact strengthening of two-disjoint-cycle-cover vertex bipancyclicity. For every ordered pair of independent edges \(e,f\in E(G)\) and every even integer \(4\leq\ell\leq N-4\), the vertex set can be partitioned into two cycles \(J_1,J_2\) of lengths \(\ell\) and \(N-\ell\), respectively, with \(e\in E(J_1)\) and \(f\in E(J_2)\), if and only if \(G\) is not isomorphic to any \(K_2\BoxProd C_{2p}\) with \(p\geq3\). The graph \(C(2,2)\cong C_4\) is treated separately: it has no 2-DCC. Consequences that retain prescribed-edge information include ordinary edge bipancyclicity and the even \(k\)-ary \(n\)-cube specialization.

发表机构

• School of Science, Minzu University of China（中央民族大学理学院）

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