Transitive path decompositions of Cartesian products of complete graphs

IF 1.4 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Designs, Codes and Cryptography Pub Date : 2024-10-03 DOI:10.1007/s10623-024-01493-9

Ajani De Vas Gunasekara, Alice Devillers

引用次数: 0

Abstract

An H-decomposition of a graph \(\Gamma \) is a partition of its edge set into subgraphs isomorphic to H. A transitive decomposition is a special kind of H-decomposition that is highly symmetrical in the sense that the subgraphs (copies of H) are preserved and transitively permuted by a group of automorphisms of \(\Gamma \). This paper concerns transitive H-decompositions of the graph \(K_n \Box K_n\) where H is a path. When n is an odd prime, we present a construction for a transitive path decomposition where the paths in the decomposition are considerably large compared to the number of vertices. Our main result supports well-known Gallai’s conjecture and an extended version of Ringel’s conjecture.

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完整图笛卡儿积的传递路径分解

一个图 \(\Gamma \)的 H 分解是将它的边集分割成与 H 同构的子图。反式分解是一种特殊的 H 分解，它具有高度对称性，即子图（H 的副本）通过 \(\Gamma \)的一组自动形变得到保留和反式置换。本文关注图 \(K_n \Box K_n\) 的传递 H 分解，其中 H 是一条路径。当 n 是奇素数时，我们提出了一种反式路径分解的构造，分解中的路径与顶点数相比相当大。我们的主要结果支持众所周知的加莱猜想和林格尔猜想的扩展版本。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.