有向循环图中的哈密顿起点与路径分解
Hamilton Starters and Path Decompositions in Directed Circulants
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中文总结 AI 辅助
针对q≥3、r≥1的有向循环图，构造了q层平衡哈密顿起点，在q=4（r≡1 mod4且r≥9）、q=3（r≡5 mod6且r足够大）时得到链兼容起点，实现弧集的最优路径分解。
中文摘要 AI 辅助
对于整数q≥3且r≥1，考虑阶为qr的循环群上的有向Cayley图，其允许步为1到r的整数。若哈密顿圈H满足：对每一个1到r的步，以及每一个模q的剩余类，H中恰好有一条该步的弧，其尾属于给定剩余类，则称H为q层平衡哈密顿起点。将H依次平移q的倍数得到的r个平移图，构成该有向图的哈密顿分解。若能从该分解的每个哈密顿圈中各选一条弧，使得所选弧构成一条简单有向路径，则称该q层平衡哈密顿起点是链兼容的。在两种构造中，兼容删除链为从0开始、到2r结束的步长为2的算术路径。当q=4时，只要r≡1 mod 4且r≥9，就可明确得到一个链兼容起点；当q=3时，对所有足够大且r≡5 mod 6的r，均存在这样的起点。证明为构造性的：四层情形使用ABAB步长字，三层情形则将有向旋转露台提升为3层平衡有向1-因子，再通过固定的四弧交换连接其两个圈。从每个平移哈密顿圈中删除唯一指定路径的弧，得到r条哈密顿路径；结合指定路径，在两种情形下均将弧集最优分解为r+1条有向路径。
英文摘要
For integers q at least 3 and r at least 1, consider the directed Cayley graph on the cyclic group of order qr whose allowed steps are the integers from 1 through r. A Hamilton cycle H is called a q-layer balanced Hamilton starter if, for every step from 1 through r and every residue class modulo q, H contains exactly one arc of that step whose tail belongs to the given residue class. The r translates of H by successive multiples of q then form a Hamilton decomposition of the digraph. A q-layer balanced Hamilton starter is called chain-compatible if one arc can be selected from each Hamilton cycle in this decomposition so that the selected arcs form a simple directed path. In both constructions, the compatible deletion chain is the arithmetic step-2 path beginning at 0 and ending at 2r. For q equal to 4, a chain-compatible starter is obtained explicitly whenever r is congruent to 1 modulo 4 and r is at least 9, while for q equal to 3 one exists for all sufficiently large r congruent to 5 modulo 6. The proofs are constructive: the four-layer case uses an ABAB step word, while in the three-layer case a directed rotational terrace is lifted to a 3-layer balanced directed 1-factor and a fixed four-arc trade joins its two cycles. Deleting the unique prescribed-path arc from each translated Hamilton cycle gives r Hamilton paths; together with the prescribed path, these form an optimal decomposition of the arc set into r+1 directed paths in both cases.
发表机构
- School of Mathematics and Statistics, Weifang University（潍坊学院数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
