the condition for a sequence to be potentially al‎,‎m‎- graphic

The set of all non-increasing non-negative integer sequences π=(d1‎,‎d2,…,dn) is denoted by nsn‎. ‎a sequence π∈nsn is said to be graphic if it is the degree sequence of a simple graph g on n vertices‎, ‎and such a graph g is called a realization of π‎. ‎the set of all graphic sequences in nsn is denoted by gsn‎. ‎the complete product split graph on l‎+‎m vertices is denoted by s¯¯¯l‎,‎m=kl∨k¯¯¯¯¯m‎, ‎where kl and km are complete graphs respectively on l=∑i=1pri and m=∑i=1psi vertices with ri and si being integers‎. ‎another split graph is denoted by sl‎,‎m=s¯¯¯r1‎,‎s1∨s¯¯¯r2‎,‎s2∨⋯∨s¯¯¯rp‎,‎sp=(kr1∨k¯¯¯¯¯s1)∨(kr2∨k¯¯¯¯¯s2)∨⋯∨(krp∨k¯¯¯¯¯sp)‎. ‎a sequence π=(d1‎,‎d2,…,dn) is said to be potentially sl‎,‎m-graphic (respectively s¯¯¯l‎,‎m)-graphic if there is a realization g of π containing sl‎,‎m (respectively s¯¯¯l‎,‎m) as a subgraph‎. ‎if π has a realization g containing sl‎,‎m on those vertices having degrees d1‎,‎d2,…,dl+m‎, ‎then π is potentially al‎,‎m-graphic‎. ‎a non-increasing sequence of non-negative integers π=(d1‎,‎d2,…,dn) is potentially al‎,‎m-graphic if and only if it is potentially sl‎,‎m-graphic‎. ‎in this paper‎, ‎we obtain the sufficient condition for a graphic sequence to be potentially al‎,‎m-graphic and this result is a generalization of that given by j‎. ‎h‎. ‎yin on split graphs‎.