The condition for a sequence to be potentially AL‎,‎M ‎- graphic

The set of all non-increasing non-negative integer sequences π=(d1ý,ýd2,…,dn) is denoted by NSný. ýA sequence π∈NSn is said to be graphic if it is the degree sequence of a simple graph G on n verticesý, ýand such a graph G is called a realization of πý. ýThe set of all graphic sequences in NSn is denoted by GSný. ýThe complete product split graph on LýýM vertices is denoted by S¯¯¯Lý,ýM=KL∨K¯¯¯¯¯Mý, ýwhere KL and KM are complete graphs respectively on L=∑i=1pri and M=∑i=1psi vertices with ri and si being integersý. ýAnother split graph is denoted by SLý,ýM=S¯¯¯r1ý,ýs1∨S¯¯¯r2ý,ýs2∨⋯∨S¯¯¯rpý,ýsp=(Kr1∨K¯¯¯¯¯s1)∨(Kr2∨K¯¯¯¯¯s2)∨⋯∨(Krp∨K¯¯¯¯¯sp)ý. ýA sequence π=(d1ý,ýd2,…,dn) is said to be potentially SLý,ýM-graphic (respectively S¯¯¯Lý,ýM)-graphic if there is a realization G of π containing SLý,ýM (respectively S¯¯¯Lý,ýM) as a subgraphý. ýIf π has a realization G containing SLý,ýM on those vertices having degrees d1ý,ýd2,…,dLý, ýthen π is potentially ALý,ýM-graphicý. ýA non-increasing sequence of non-negative integers π=(d1ý,ýd2,…,dn) is potentially ALý,ýM-graphic if and only if it is potentially SLý,ýM-graphicý. ýIn this paperý, ýwe obtain the sufficient condition for a graphic sequence to be potentially ALý,ýM -graphic and this result is a generalization of that given by Jý. ýHý. ýYin on split graphsý.