Ordered and colored subgraph density problems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911801197723648 |
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| author | Cairncross, Emily Mubayi, Dhruv |
| author_facet | Cairncross, Emily Mubayi, Dhruv |
| contents | We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of $k$-edge stars in a graph with density $x \in [0, 1]$ is asymptotically maximized by a clique and isolated vertices or its complement. Next, among ordered $n$-vertex graphs with $m$ edges, we determine the maximum and minimum number of copies of a $k$-edge star whose nonleaf vertex is minimum among all vertices of the star. Finally, for $s \ge 2$, we define a particular $3$-edge-colored complete graph $F$ on $2s$ vertices with colors blue, green and red, and determine, for each $(x_b, x_g)$ with $x_b+x_g\le 1$ and $x_b, x_g \ge 0$, the maximum density of $F$ in a large graph whose blue, green and red edge sets have densities $x_b, x_g$ and $1-x_b-x_g$, respectively. These are the first nontrivial examples of colored graphs for which such complete results are proved. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12016 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ordered and colored subgraph density problems Cairncross, Emily Mubayi, Dhruv Combinatorics We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of $k$-edge stars in a graph with density $x \in [0, 1]$ is asymptotically maximized by a clique and isolated vertices or its complement. Next, among ordered $n$-vertex graphs with $m$ edges, we determine the maximum and minimum number of copies of a $k$-edge star whose nonleaf vertex is minimum among all vertices of the star. Finally, for $s \ge 2$, we define a particular $3$-edge-colored complete graph $F$ on $2s$ vertices with colors blue, green and red, and determine, for each $(x_b, x_g)$ with $x_b+x_g\le 1$ and $x_b, x_g \ge 0$, the maximum density of $F$ in a large graph whose blue, green and red edge sets have densities $x_b, x_g$ and $1-x_b-x_g$, respectively. These are the first nontrivial examples of colored graphs for which such complete results are proved. |
| title | Ordered and colored subgraph density problems |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.12016 |