On potentially $K_{r+1}-U$-graphical Sequences

Lai Chunhui Minnan Normal University Yan Guiying Academy of Mathematics and Systems Science

Combinatorics mathscidoc:2402.06004

Utilitas Mathematica, 80, 233-244, 2009.11
Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-U, n)$ for $n\geq 5r+18, r+1 \geq k \geq 7,$ $j \geq 6$ where $U$ is a graph on $k$ vertices and $j$ edges which contains a graph $K_3 \bigcup P_3$ but not contains a cycle on $4$ vertices and not contains $Z_4$.
graph; degree sequence; potentially $K_{r+1}-U$-graphic sequence; potentially $K_{r+1}-K_3 \bigcup P_3$-graphic sequence
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  title={On potentially $K_{r+1}-U$-graphical Sequences},
  author={Lai Chunhui, and Yan Guiying},
  booktitle={Utilitas Mathematica},
Lai Chunhui, and Yan Guiying. On potentially $K_{r+1}-U$-graphical Sequences. 2009. Vol. 80. In Utilitas Mathematica. pp.233-244. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20240212155833880918767.
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