Anal. Theory Appl., 34 (2018), pp. 187-198.

Published online: 2018-07

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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.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2018.v34.n2.8}, url = {http://global-sci.org/intro/article_detail/ata/12586.html} }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.

*Analysis in Theory and Applications*.

*34*(2). 187-198. doi:10.4208/ata.2018.v34.n2.8