The condition for a sequence to be potentially A_{L‎, ‎M}‎- graphic

The set of all non-increasing non-negative integer sequences \pi=(d_۱‎, ‎d_۲,\ldots,d_n) is denoted by NS_n‎. ‎A sequence \pi\in NS_{n} 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 \pi‎. ‎The set of all graphic sequences in NS_{n} is denoted by GS_{n}‎. ‎The complete product split graph on L‎ + ‎M vertices is denoted by \overline{S}_{L‎, ‎M}=K_{L} \vee \overline{K}_{M}‎, ‎where K_{L} and K_{M} are complete graphs respectively on L = \sum\limits_{i = ۱}^{p}r_{i} and M = \sum\limits_{i = ۱}^{p}s_{i} vertices with r_{i} and s_{i} being integers‎. ‎Another split graph is denoted by S_{L‎, ‎M} = \overline{S}_{r_{۱}‎, ‎s_{۱}} \vee\overline{S}_{r_{۲}‎, ‎s_{۲}} \vee \cdots \vee \overline{S}_{r_{p}‎, ‎s_{p}}= (K_{r_{۱}} \vee \overline{K}_{s_{۱}})\vee (K_{r_{۲}} \vee \overline{K}_{s_{۲}})\vee \cdots \vee (K_{r_{p}} \vee \overline{K}_{s_{p}})‎. ‎A sequence \pi=(d_{۱}‎, ‎d_{۲},\ldots,d_{n}) is said to be potentially S_{L‎, ‎M}-graphic (respectively \overline{S}_{L‎, ‎M})-graphic if there is a realization G of \pi containing S_{L‎, ‎M} (respectively \overline{S}_{L‎, ‎M}) as a subgraph‎. ‎If \pi has a realization G containing S_{L‎, ‎M} on those vertices having degrees d_{۱}‎, ‎d_{۲},\ldots,d_{L+M}‎, ‎then \pi is potentially A_{L‎, ‎M}-graphic‎. ‎A non-increasing sequence of non-negative integers \pi = (d_{۱}‎, ‎d_{۲},\ldots,d_{n}) is potentially A_{L‎, ‎M}-graphic if and only if it is potentially S_{L‎, ‎M}-graphic‎. ‎In this paper‎, ‎we obtain the sufficient condition for a graphic sequence to be potentially A_{L‎, ‎M}-graphic and this result is a generalization of that given by J‎. ‎H‎. ‎Yin on split graphs‎.

کلیدواژه ها:

‎Split graph‎ ، ‎complete product split graph‎ ، ‎potentially H-graphic Sequences