Volume 34, Issue 2
On Potentially Graphical Sequences of $G−E(H)$

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

Published online: 2018-07

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• Abstract

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.

• Keywords

Graph, $(a,b,n)$-graph, potentially graphical sequences.

05C07

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@Article{ATA-34-187, author = {}, title = {On Potentially Graphical Sequences of $G−E(H)$}, journal = {Analysis in Theory and Applications}, year = {2018}, volume = {34}, number = {2}, pages = {187--198}, abstract = {

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} }
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.

Bilal A. Chat & S. Pirzada. (1970). On Potentially Graphical Sequences of $G−E(H)$. Analysis in Theory and Applications. 34 (2). 187-198. doi:10.4208/ata.2018.v34.n2.8
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