TY - JOUR T1 - On Potentially Graphical Sequences of $G−E(H)$ JO - Analysis in Theory and Applications VL - 2 SP - 187 EP - 198 PY - 2018 DA - 2018/07 SN - 34 DO - http://doi.org/10.4208/ata.2018.v34.n2.8 UR - https://global-sci.org/intro/article_detail/ata/12586.html KW - Graph, $(a,b,n)$-graph, potentially graphical sequences. AB -

A loopless graph on $n$ vertices in which vertices are connected at least by $a$ and at most by $b$ edges is called a $(a,b,n)$-graph. A $(b,b,n)$-graph is called $(b,n)$-graph and is denoted by $K^b_n$ (it is a complete graph), its complement by $\overline{K}^b_n$. A non increasing sequence $π = (d_1,···,d_n)$ of nonnegative integers is said to be $(a,b,n)$ graphic if it is realizable by an $(a,b,n)$-graph. We say a simple graphic sequence $π = (d_1,···,d_n)$ is potentially $K_4−K_2\cup K_2$-graphic if it has a a realization containing an $K_4−K_2\cup K_2$ as a subgraph where $K_4$ is a complete graph on four vertices and $K_2\cup K_2$ is a set of independent edges. In this paper, we find the smallest degree sum such that every $n$-term graphical sequence contains $K_4−K_2\cup K_2$ as subgraph.