Graphic sequences with a realization containing a generalized friendship graph

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 HH, determine the smallest even integer σ(H,n)σ(H,n) such that every nn-term graphic sequence π=(d1,d2,…,dn)π=(d1,d2,…,dn) with term sum σ(π)=d1+d2+⋯+dn≥σ(H,n)σ(π)=d1+d2+⋯+dn≥σ(H,n) has a realization GG containing HH as a subgraph. Let Ft,r,kFt,r,k denote the generalized friendship graph on kt−kr+rkt−kr+r vertices, that is, the graph of kk copies of KtKt meeting in a common rr set, where KtKt is the complete graph on tt vertices and 0≤r≤t0≤r≤t. In this paper, we determine σ(Ft,r,k,n)σ(Ft,r,k,n) for k≥2k≥2, t≥3t≥3, 1≤r≤t−21≤r≤t−2 and nn sufficiently large.