TY - JOUR
ID - 20361
TI - The condition for a sequence to be potentially $A_{L, M}$- graphic
JO - Transactions on Combinatorics
JA - TOC
LA - en
SN - 2251-8657
AU - Pirzada, Shariefuddin
AU - A. Chat, Bilal
AD - University of Kashmir
Y1 - 2017
PY - 2017
VL - 6
IS - 1
SP - 21
EP - 27
KW - Split graph
KW - complete product split graph
KW - potentially $H$-graphic Sequences
DO - 10.22108/toc.2017.20361
N2 - The set of all non-increasing non-negative integer sequences $pi=(d_1, d_2,ldots,d_n)$ is denoted by $NS_n$. A sequence $piin NS_{n}$ 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 $L = sumlimits_{i = 1}^{p}r_{i}$ and $M = sumlimits_{i = 1}^{p}s_{i}$ vertices with $r_{i}$ and $s_{i}$ being integers. Another split graph is denoted by $S_{L, M} = overline{S}_{r_{1}, s_{1}} veeoverline{S}_{r_{2}, s_{2}} vee cdots vee overline{S}_{r_{p}, s_{p}}= (K_{r_{1}} vee overline{K}_{s_{1}})vee (K_{r_{2}} vee overline{K}_{s_{2}})vee cdots vee (K_{r_{p}} vee overline{K}_{s_{p}})$. A sequence $pi=(d_{1}, d_{2},ldots,d_{n})$ 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 $d_{1}, d_{2},ldots,d_{L+M}$, then $pi$ is potentially $A_{L, M}$-graphic. A non-increasing sequence of non-negative integers $pi = (d_{1}, d_{2},ldots,d_{n})$ 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.
UR - https://toc.ui.ac.ir/article_20361.html
L1 - https://toc.ui.ac.ir/article_20361_5539a345ae0f45bb6974e8e9397a9145.pdf
ER -