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Extremal Graphs for Blow-Ups of Keyrings
Graphs and Combinatorics ( IF 0.6 ) Pub Date : 2020-07-01 , DOI: 10.1007/s00373-020-02203-7
Zhenyu Ni , Liying Kang , Erfang Shan , Hui Zhu

The blow-up of a graph H is the graph obtained from replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different. Given a graph H and a positive integer n, the extremal number, ex(nH), is the maximum number of edges in a graph on n vertices that does not contain H as a subgraph. A keyring \(C_s(k)\) is a \((k+s)\)-edge graph obtained from a cycle of length k by appending s leaves to one of its vertices. This paper determines the extremal number and finds the extremal graphs for the blow-ups of keyrings \(C_s(k)\) (\(k\ge 3\), \(s\ge 1\)) when n is sufficiently large. For special cases when \(k=0\) or \(s=0\), the extremal number of the blow-ups of the graph \(C_s(0)\) (a star) has been determined by Erdös et al. (J Comb Theory Ser B 64:89–100, 1995) and Chen et al. (J Comb Theory Ser B 89: 159–171, 2003), while the extremal number and extremal graphs for the blow-ups of the graph \(C_0(k)\) (a cycle) when n is sufficiently large has been determined by Liu (Electron J Combin 20: P65, 2013).



中文翻译:

钥匙圈爆破的极端图

H的爆炸是通过用相同大小的集团替换H中的每个边缘而获得的图,其中集团的新顶点都不同。给定一个图H和一个正整数n,极值ex(n,  H)是n个不包含H作为子图的顶点在图上的最大边数。密钥环\(C_s(k)\)\((k + s)\)-边图,是通过将s附加到长度为k的循环而获得离开其顶点之一。当n足够大时,本文确定了极值数量并找到了密匙环\(C_s(k)\)\(k \ ge 3 \)\(s \ ge 1 \))爆炸的极值图。对于\(k = 0 \)\(s = 0 \)的特殊情况,Erdös等人已经确定了图\(C_s(0)\)(星形)爆炸的极值数目。(J Comb Theory Ser B 64:89-100,1995)和Chen等。(J Comb Theory Ser B 89:159–171,2003),而当n为n时,图\(C_0(k)\)(一个周期)爆炸的极值数和极值图。 Liu已确定其足够大(Electron J Combin 20:P65,2013)。

更新日期:2020-07-01
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