(untitled) (Analysis in Theory and Applications, 2018)
A loopless graph on n vertices in which vertices are connected at least by a and at most by b edges is called a (a,b,n)-graph. A (b,b,n)-graph is called (b,n)-graph and is denoted by K^b_n (it is a complete graph), its complement by K^b_n. A non increasing sequence π = (d_1,···,d_n) of nonnegative integers is said to be (a,b,n) graphic if it is realizable by an (a,b,n)-graph. We say a simple graphic sequence π = (d_1,···,d_n) is potentially K_4−K_2K_2-graphic if it has a a realization containing an K_4−K_2K_2 as a subgraph where K_4 is a complete graph on four vertices and K_2K_2 is a set of independent edges. In this paper, we find the smallest degree sum such that every n-term graphical sequence contains K_4−K_2K_2 as subgraph.