A note on the multicolor size-Ramsey numbers of connected graphs
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910113963442176 |
|---|---|
| author | DeBiasio, Louis |
| author_facet | DeBiasio, Louis |
| contents | The $r$-color size-Ramsey number of a graph $H$, denoted by $\widehat{R}_r(H)$, is the minimum number of edges in a graph $G$ having the property that every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H$.
Krivelevich proved that $\widehat{R}_r(P_{m+1})=Ω(r^2m)$ where $P_{m+1}$ is the path on $m$ edges. He explains that his proof actually applies to any connected graph $H$ with $m$ edges and vertex cover number larger than $\sqrt{m}$. He also notes that some restriction on the vertex cover number is necessary since the star with $m$ edges, $K_{1,m}$, has vertex cover number 1 and satisfies $\widehat{R}_r(K_{1,m})=r(m-1)+1$. We prove that the star is actually the only exception; that is, $\widehat{R}_r(H)=Ω(r^2m)$ for every non-star connected graph $H$ with $m$ edges.
We also prove a strengthening of this result for trees. It follows from results of Beck and Dellamonica that $\widehat{R}_2(T)=Θ(β(T))$ for every tree $T$ with bipartition $\{V_1, V_2\}$ and $β(T)=|V_1|\max\{d(v):v\in V_1\}+|V_2|\max\{d(v):v\in V_2\}$. We prove that $\widehat{R}_r(T)=Ω(r^2β(T))$ for every tree $T$, again with the exception of the star. Additionally, we prove that for the family of non-star trees $T$ with $β(T)=Ω(n_1n_2)$ (which includes all non-star trees of linear maximum degree and all trees of radius 2 for example) we have $\widehat{R}_r(T)=Θ(r^2β(T))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05856 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on the multicolor size-Ramsey numbers of connected graphs DeBiasio, Louis Combinatorics The $r$-color size-Ramsey number of a graph $H$, denoted by $\widehat{R}_r(H)$, is the minimum number of edges in a graph $G$ having the property that every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H$. Krivelevich proved that $\widehat{R}_r(P_{m+1})=Ω(r^2m)$ where $P_{m+1}$ is the path on $m$ edges. He explains that his proof actually applies to any connected graph $H$ with $m$ edges and vertex cover number larger than $\sqrt{m}$. He also notes that some restriction on the vertex cover number is necessary since the star with $m$ edges, $K_{1,m}$, has vertex cover number 1 and satisfies $\widehat{R}_r(K_{1,m})=r(m-1)+1$. We prove that the star is actually the only exception; that is, $\widehat{R}_r(H)=Ω(r^2m)$ for every non-star connected graph $H$ with $m$ edges. We also prove a strengthening of this result for trees. It follows from results of Beck and Dellamonica that $\widehat{R}_2(T)=Θ(β(T))$ for every tree $T$ with bipartition $\{V_1, V_2\}$ and $β(T)=|V_1|\max\{d(v):v\in V_1\}+|V_2|\max\{d(v):v\in V_2\}$. We prove that $\widehat{R}_r(T)=Ω(r^2β(T))$ for every tree $T$, again with the exception of the star. Additionally, we prove that for the family of non-star trees $T$ with $β(T)=Ω(n_1n_2)$ (which includes all non-star trees of linear maximum degree and all trees of radius 2 for example) we have $\widehat{R}_r(T)=Θ(r^2β(T))$. |
| title | A note on the multicolor size-Ramsey numbers of connected graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.05856 |