Graphic sequences with a realization containing a generalized friendship graph - Yin, Jian-Hua; Chen, Gang; Schmitt, John R. (Discrete mathematics, 2008)
Gould, Jacobson and Lehel [R.J. Gould, M.S. Jacobson, J. Lehel, Potentially G-graphical degree sequences, in: Y. Alavi, et al. (Eds.), Combinatorics, Graph Theory and Algorithms, vol. I, New Issues Press, Kalamazoo, MI, 1999, pp. 451–460] considered a variation of the classical Turán-type extremal problems as follows: for any simple graph H , determine the smallest even integer σ ( H , n ) such that every n -term graphic sequence π = ( d 1 , d 2 , … , d n ) with term sum σ ( π ) = d 1 + d 2 + ⋯ + d n ≥ σ ( H , n ) has a realization G containing H as a subgraph. Let F t , r , k denote the generalized friendship graph on k t − k r + r vertices, that is, the graph of k copies of K t meeting in a common r set, where K t is the complete graph on t vertices and 0 ≤ r ≤ t . In this paper, we determine σ ( F t , r , k , n ) for k ≥ 2 , t ≥ 3 , 1 ≤ r ≤ t − 2 and n sufficiently large.