Abstract
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(n, H), 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).
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Research was partially supported by NSFC (Grant numbers 11971298, 11871329).
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Ni, Z., Kang, L., Shan, E. et al. Extremal Graphs for Blow-Ups of Keyrings. Graphs and Combinatorics 36, 1827–1853 (2020). https://doi.org/10.1007/s00373-020-02203-7
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DOI: https://doi.org/10.1007/s00373-020-02203-7