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

Lai Chunhui Minnan Normal University

Combinatorics mathscidoc:2402.06005

Ars Combinatoria, 102, 65 – 77, 2011.10
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 $\sigma (K_{r+1}-Z, n)$ 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 $n\geq 5r+16, r\geq 4$.
subgraph; degree sequence; potentially $K_{r+1}-Z$-graphic; potentially $K_{r+1}-Z_4$-graphic sequence
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  title={The smallest degree sum that yields potentially $K_{r+1}-Z$-graphical Sequences},
  author={Lai Chunhui},
  booktitle={Ars Combinatoria},
  pages={65 – 77},
Lai Chunhui. The smallest degree sum that yields potentially $K_{r+1}-Z$-graphical Sequences. 2011. Vol. 102. In Ars Combinatoria. pp.65 – 77.
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