THE CONDITION FOR A SEQUENCE TO BE POTENTIALLY AL;M- GRAPHIC

The set of all non-increasing non-negative integer sequences p = (d1; d2; : : : ; dn) is denoted by NSn. A sequence p 2 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 p. The set of all graphic sequences in NSn is denoted by GSn. The complete product split graph on L+M vertices is denoted by SL;M = KL _KM, where KL and KM are complete graphs respectively on L = Σp i=1 ri and M = Σp i=1 si ertices with ri and si being integers. Another split graph is denoted by SL;M = Sr1;s1 _ Sr2;s2 _ _ Srp;sp = (Kr1 _ Ks1 ) _ (Kr2 _ Ks2 ) _ _ (Krp _ Ksp ). A sequence = (d1; d2; : : : ; dn) is said to be po- tentially SL;M-graphic (respectively SL;M)-graphic if there is a realization G of containing SL;M (respectively SL;M) as a subgraph. If has a realization G containing SL;M on those vertices having degrees d1; d2; : : : ; dL+M, then p is potentially AL;M-graphic. A non-increasing sequence of non-negative integers p = (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.