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An extremal problem for a graphic sequence to have a realization containing every 2-tree with prescribed size
| Content Provider | Hyper Articles en Ligne (HAL) |
|---|---|
| Author | Zeng, De-Yan Yin, Jian-Hua |
| Abstract | A graph $G$ is a $2$-tree if $G=K_3$, or $G$ has a vertex $v$ of degree 2, whose neighbors are adjacent, and $G-v$ is a 2-tree. Clearly, if $G$ is a 2-tree on $n$ vertices, then $|E(G)|=2n-3$. A non-increasing sequence $\pi =(d_1, \ldots ,d_n)$ of nonnegative integers is a graphic sequence if it is realizable by a simple graph $G$ on $n$ vertices. Yin and Li (Acta Mathematica Sinica, English Series, 25(2009)795–802) proved that if $k \geq 2$, $n \geq \frac{9}{2}k^2 + \frac{19}{2}k$ and $\pi =(d_1, \ldots ,d_n)$ is a graphic sequence with $\sum \limits_{i=1}^n d_i > (k-2)n$, then $\pi$ has a realization containing every tree on $k$ vertices as a subgraph. Moreover, the lower bound $(k-2)n$ is the best possible. This is a variation of a conjecture due to Erdős and Sós. In this paper, we investigate an analogue extremal problem for 2-trees and prove that if $k \geq 3$, $n \geq 2k^2-k$ and $\pi =(d_1, \ldots ,d_n)$ is a graphic sequence with $\sum \limits_{i=1}^n d_i > \frac{4kn}{3} - \frac{5n}{3}$ then $\pi$ has a realization containing every 2-tree on $k$ vertices as a subgraph. We also show that the lower bound $\frac{4kn}{3} - \frac{5n}{3}$ is almost the best possible. |
| Ending Page | 326 |
| Page Count | 12 |
| Starting Page | 315 |
| File Format | |
| ISSN | 14627264 |
| e-ISSN | 13658050 |
| Journal | Discrete Mathematics and Theoretical Computer Science |
| Issue Number | 3 |
| Language | English |
| Publisher | DMTCS |
| Publisher Date | 2016-08-06 |
| Access Restriction | Open |
| Subject Keyword | 2-trees. graphic sequences realization degree sequences info Computer Science [cs] Discrete Mathematics [cs.DM] |
| Content Type | Text |
| Resource Type | Article |
| Subject | Discrete Mathematics and Combinatorics Theoretical Computer Science Computer Science |
