# The smallest degree sum that yields potentially $K_{r+1}-Z$-graphical Sequences

• 作者： Chunhui LAI 1
• 作者单位：
• 通讯作者： Chunhui LAI  Email:laichunhui@mnnu.edu.cn
• 提交时间：2024-02-13

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 $\sigma (K_{r+1}-Z, n)$ for
$n\geq 5r+19, r+1 \geq k \geq 5,$  $j \geq 5$ where $Z$ is a graph on $k$
vertices and $j$ edges which
contains a graph  $Z_4$  but
not contains a cycle on $4$ vertices. We also determine the values of
$\sigma (K_{r+1}-Z_4, n)$, $\sigma (K_{r+1}-(K_4-e), n)$,
$\sigma (K_{r+1}-K_4, n)$ for
$n\geq 5r+16, r\geq 4$.

#### 版本历史

 [V1] 2024-02-13 20:00:00 ChinaXiv:202402.00146V1

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