Graphs and Combinatorics ( IF 0.6 ) Pub Date : 2021-03-25 , DOI: 10.1007/s00373-021-02301-0 J. Leaños , Christophe Ndjatchi
Let G be a simple graph of order \(n\ge 2\) and let \(k\in \{1,\ldots ,n-1\}\). The k-token graph \(F_k(G)\) of G is the graph whose vertices are the k-subsets of V(G), where two vertices are adjacent in \(F_k(G)\) whenever their symmetric difference is an edge of G. In 2018 Leaños and Trujillo-Negrete proved that if G is t-connected and \(t\ge k\), then \(F_k(G)\) is at least \(k(t-k+1)\)-connected. In this paper we show that such a lower bound remains true in the context of edge-connectivity. Specifically, we show that if G is t-edge-connected and \(t\ge k\), then \(F_k(G)\) is at least \(k(t-k+1)\)-edge-connected. We also provide some families of graphs attaining this bound.
中文翻译:
令牌图的边缘连通性
令G为\(n \ ge 2 \)阶的简单图,并令\(k \ in \ {1,\ ldots,n-1 \} \)。所述ķ -token图表\(F_k(G)\)的ģ是其顶点是图ķ的-subsets V(G ^),其中两个顶点在彼此相邻\(F_k(G)\)每当其对称差是G的边缘。在2018年,Leaños和Trujillo-Negrete证明,如果G是t连通的且\(t \ ge k \),则\(F_k(G)\)至少是\(k(t-k + 1)\)-连接的。在本文中,我们表明,在边缘连接的情况下,这样的下限仍然适用。具体来说,我们证明如果G是t -edge-connected且\(t \ ge k \),则\(F_k(G)\)至少是\(k(t-k + 1)\)- edge-连接的。我们还提供了一些达到此限制的图形族。