Locally triangular graphs and rectagraphs with symmetry
Locally triangular graphs and rectagraphs with symmetry
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DOI：
10.1016/j.jcta.2015.01.006
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发表时间：
2014-07
期刊：
影响因子：
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通讯作者：
J. Bamberg;Alice Devillers;Joanna B. Fawcett;C. Praeger
中科院分区：
文献类型：
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作者：
J. Bamberg;Alice Devillers;Joanna B. Fawcett;C. Praeger
已知局部三角形图是二分直角图的减半图，它是无三角形连通图，其中每条 2 弧都位于一个唯一的四边形中。如果存在 G⩽ Aut (Γ)，使得对于每个顶点 u，由顶点稳定器 Gu 在邻域 Γ (u) 上导出的置换群是秩 3 传递的，则图 Γ 是局部秩 3。寻找局部秩 3 图的一个自然位置是在局部三角图中，其中每个导出的邻域图与三角图 T n 同构。这是因为图 T n 的顶点集为 {1,…, n} 的 2-子集，边集为相交于一点的 2-子集对，它承认 3 阶自同构群。在本文中，我们对某个直角图族进行了分类，对于该直角图族，由 Aut (Г) u 在 Г (u) 上导出的置换群对于某些顶点 u 是 4-齐次的。然后，我们使用这个结果对局部连通的三角形图进行分类，这些三角形图的局部等级也为 3。
Locally triangular graphs are known to be halved graphs of bipartite rectagraphs, which are connected triangle-free graphs in which every 2-arc lies in a unique quadrangle. A graph Γ is locally rank 3 if there exists G⩽ Aut (Γ) such that for each vertex u, the permutation group induced by the vertex stabiliser G u on the neighbourhood Γ (u) is transitive of rank 3. One natural place to seek locally rank 3 graphs is among the locally triangular graphs, where every induced neighbourhood graph is isomorphic to a triangular graph T n. This is because the graph T n, which has vertex set the 2-subsets of {1,…, n} and edge set the pairs of 2-subsets intersecting at one point, admits a rank 3 group of automorphisms. In this paper, we classify a certain family of rectagraphs for which the permutation group induced by Aut (Γ) u on Γ (u) is 4-homogeneous for some vertex u. We then use this result to classify the connected locally triangular graphs that are also locally rank 3.