Ramsey Number Counterexample Checking and One Vertex Extension Linearly Bound by $s$ and $t$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916496955932672 |
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| author | Lehavi, Adam M. |
| author_facet | Lehavi, Adam M. |
| contents | The Ramsey number $R(s,t)$ is the smallest integer $n$ such that all graphs of size $n$ contain a clique of size $s$ or an independent set of size $t$. $\mathcal{R}(s,t,n)$ is the set of all counterexample graphs without this property for a given $n$. We prove that if a graph $G_{n+1}$ of size $n+1$ has $\max\{s,t\}+1$ subgraphs in $\mathcal{R}(s,t,n)$, then $G_{n+1}$ is in $\mathcal{R}(s,t,n+1)$. Based on this, we introduce algorithms for one-vertex extension and counterexample checking with runtime linearly bound by $s$ and $t$. We prove the utility of these algorithms by verifying $\mathcal{R}(4,6,36)$ and $\mathcal{R}(5,5,43)$ are empty given current sets $\mathcal{R}(4,6,35)$ and $\mathcal{R}(5,5,42)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_04267 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ramsey Number Counterexample Checking and One Vertex Extension Linearly Bound by $s$ and $t$ Lehavi, Adam M. Combinatorics 05D10 (Primary), 05C55 (Secondary) The Ramsey number $R(s,t)$ is the smallest integer $n$ such that all graphs of size $n$ contain a clique of size $s$ or an independent set of size $t$. $\mathcal{R}(s,t,n)$ is the set of all counterexample graphs without this property for a given $n$. We prove that if a graph $G_{n+1}$ of size $n+1$ has $\max\{s,t\}+1$ subgraphs in $\mathcal{R}(s,t,n)$, then $G_{n+1}$ is in $\mathcal{R}(s,t,n+1)$. Based on this, we introduce algorithms for one-vertex extension and counterexample checking with runtime linearly bound by $s$ and $t$. We prove the utility of these algorithms by verifying $\mathcal{R}(4,6,36)$ and $\mathcal{R}(5,5,43)$ are empty given current sets $\mathcal{R}(4,6,35)$ and $\mathcal{R}(5,5,42)$. |
| title | Ramsey Number Counterexample Checking and One Vertex Extension Linearly Bound by $s$ and $t$ |
| topic | Combinatorics 05D10 (Primary), 05C55 (Secondary) |
| url | https://arxiv.org/abs/2411.04267 |