Bibliographic Details
| Title: |
Many Triangles with Few Edges |
| Authors: |
Kirsch, R., Radcliffe, A. J. |
| Source: |
Electronic Journal of Combinatorics, Volume 26, Issue 2 (2019), P2.36 |
| Publication Year: |
2017 |
| Collection: |
Mathematics |
| Subject Terms: |
Mathematics - Combinatorics |
| More Details: |
Extremal problems concerning the number of independent sets or complete subgraphs in a graph have been well studied in recent years. Cutler and Radcliffe proved that among graphs with $n$ vertices and maximum degree at most $r$, where $n = a(r+1)+b$ and $0 \le b \le r$, $aK_{r+1}\cup K_b$ has the maximum number of complete subgraphs, answering a question of Galvin. Gan, Loh, and Sudakov conjectured that $aK_{r+1}\cup K_b$ also maximizes the number of complete subgraphs $K_t$ for each fixed size $t \ge 3$, and proved this for $a = 1$. Cutler and Radcliffe proved this conjecture for $r \le 6$. We investigate a variant of this problem where we fix the number of edges instead of the number of vertices. We prove that $aK_{r+1}\cup \mathcal{C}(b)$, where $\mathcal{C}(b)$ is the colex graph on $b$ edges, maximizes the number of triangles among graphs with $m$ edges and any fixed maximum degree $r\le 8$, where $m = a \binom{r+1}{2} + b$ and $0 \le b < \binom{r+1}{2}$. |
| Document Type: |
Working Paper |
| Access URL: |
http://arxiv.org/abs/1709.06163 |
| Accession Number: |
edsarx.1709.06163 |
| Database: |
arXiv |
