A new framework for identifying most reliable graphs and a correction to the $K_{3,3}$-theorem
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arXiv
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| Format: | Preprint |
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2024
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| author | Landgren, Lorents F. Steif, Jeffrey E. |
| author_facet | Landgren, Lorents F. Steif, Jeffrey E. |
| contents | Given a multigraph $G$, the all-terminal reliability $R(G,p)$ is the probability that $G$ remains connected under percolation with parameter $p$. Fixing the number of vertices $n$ and edges $m$, we investigate which graphs maximize $R(G,p)$ -- such graphs are called optimal -- paying particular attention to uniqueness and to whether the answer depends upon $p$. We generalize the concept of a distillation and build a framework with which we identify all optimal graphs for which $m-n\in\{1,2,3\}$. These graphs are uniformly optimal in $p$. Most have been previously identified, but with serious problems, especially when $m-n=3$. We obtain partial results for $m-n\in\{4,5\}$.
For $m-n=3$, the optimal graphs were incorrectly identified by Wang in 1994, in the infinite number of cases where $m\equiv5\pmod{9}$ and $m\geq14$. This erroneous result concerns subdivisions of $K_{3,3}$ and has been cited extensively, without any mistake being detected. While the optimal graphs were correctly described for other $m$, the proof is fundamentally flawed. Our proof of the rectified statement is self-contained.
For $m-n=4$, the optimal graphs were recently shown to depend upon $p$ for infinitely many $m$. We find a new such set of $m$-values, which gives a different perspective on why this phenomenon occurs and leads us to conjecture that uniformly optimal graphs exist only for finitely many $m$. However, for $m-n=5$, we conjecture that there are again infinitely many uniformly optimal graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20217 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A new framework for identifying most reliable graphs and a correction to the $K_{3,3}$-theorem Landgren, Lorents F. Steif, Jeffrey E. Combinatorics Probability 05Cxx, 82B43, 60C05 Given a multigraph $G$, the all-terminal reliability $R(G,p)$ is the probability that $G$ remains connected under percolation with parameter $p$. Fixing the number of vertices $n$ and edges $m$, we investigate which graphs maximize $R(G,p)$ -- such graphs are called optimal -- paying particular attention to uniqueness and to whether the answer depends upon $p$. We generalize the concept of a distillation and build a framework with which we identify all optimal graphs for which $m-n\in\{1,2,3\}$. These graphs are uniformly optimal in $p$. Most have been previously identified, but with serious problems, especially when $m-n=3$. We obtain partial results for $m-n\in\{4,5\}$. For $m-n=3$, the optimal graphs were incorrectly identified by Wang in 1994, in the infinite number of cases where $m\equiv5\pmod{9}$ and $m\geq14$. This erroneous result concerns subdivisions of $K_{3,3}$ and has been cited extensively, without any mistake being detected. While the optimal graphs were correctly described for other $m$, the proof is fundamentally flawed. Our proof of the rectified statement is self-contained. For $m-n=4$, the optimal graphs were recently shown to depend upon $p$ for infinitely many $m$. We find a new such set of $m$-values, which gives a different perspective on why this phenomenon occurs and leads us to conjecture that uniformly optimal graphs exist only for finitely many $m$. However, for $m-n=5$, we conjecture that there are again infinitely many uniformly optimal graphs. |
| title | A new framework for identifying most reliable graphs and a correction to the $K_{3,3}$-theorem |
| topic | Combinatorics Probability 05Cxx, 82B43, 60C05 |
| url | https://arxiv.org/abs/2407.20217 |