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Neighbour-transitive codes in Kneser graphs
Journal of Combinatorial Theory Series A ( IF 1.1 ) Pub Date : 2023-12-14 , DOI: 10.1016/j.jcta.2023.105850
Dean Crnković , Daniel R. Hawtin , Nina Mostarac , Andrea Švob

A code C is a subset of the vertex set of a graph and C is s-neighbour-transitive if its automorphism group Aut(C) acts transitively on each of the first s+1 parts C0,C1,,Cs of the distance partition {C=C0,C1,,Cρ}, where ρ is the covering radius of C. While codes have traditionally been studied in the Hamming and Johnson graphs, we consider here codes in the Kneser graphs. Let Ω be the underlying set on which the Kneser graph K(n,k) is defined. Our first main result says that if C is a 2-neighbour-transitive code in K(n,k) such that C has minimum distance at least 5, then n=2k+1 (i.e., C is a code in an odd graph) and C lies in a particular infinite family or is one particular sporadic example. We then prove several results when C is a neighbour-transitive code in the Kneser graph K(n,k). First, if Aut(C) acts intransitively on Ω we characterise C in terms of certain parameters. We then assume that Aut(C) acts transitively on Ω, first proving that if C has minimum distance at least 3 then either K(n,k) is an odd graph or Aut(C) has a 2-homogeneous (and hence primitive) action on Ω. We then assume that C is a code in an odd graph and Aut(C) acts imprimitively on Ω and characterise C in terms of certain parameters. We give examples in each of these cases and pose several open problems.



中文翻译:


克内泽图中的邻域传递码



代码 C 是图的顶点集的子集,并且如果其自同构群 Aut(C) 传递地作用于每个前 s+1 部分 C0,C1,,Cs 的 ,其中 ρ 是 C 的覆盖半径。虽然传统上是在 Hamming 和 Johnson 图中研究代码,但我们在这里考虑 Kneser 图中的代码。令 Ω 为定义克尼泽图 K(n,k) 的基础集。我们的第一个主要结果表明,如果 C 是 K(n,k) 中的 2 邻域传递代码,使得 C 的最小距离至少为 5,则 n=2k+1 (即,C 是奇数图)并且 C 位于特定的无限族中或者是一个特定的零星示例。然后,我们证明当 C 是克尼塞尔图 K(n,k) 中的邻传递码时的几个结果。首先,如果 Aut(C) 不及物地作用于Ω,我们就用某些参数来表征C。然后我们假设 Aut(C) 传递地作用于 Ω,首先证明如果 C 的最小距离至少为 3,则 K(n,k) 是奇数图,或者 Aut(C) 具有2-对 Ω 的同质(因此原始)作用。然后,我们假设 C 是奇数图中的代码,并且 Aut(C) 直接作用于 Ω 并根据某些参数来表征 C。我们对每种情况都给出了例子,并提出了几个未解决的问题。

更新日期:2023-12-15
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