A Simple Proof of the Gross-Saccoman Multigraph Conjecture
Resumen:
An enigmatic conjecture in network synthesis asserts that the the uniformly most reliable multigraphs are simple. Daniel Gross and John Saccoman proved in 1998 that the answer is affirmative whenever m ≤ n + 2, where n and m is the respective number of nodes and edges of the multigraphs. They conjectured that the optimality is also achieved by simple graphs when m = n + 3. A proof for this conjecture recently appeared. In this article we provide a unified short proof for the previous cases where m ≤ n + 3. Our proof strategy holds whenever the most reliable simple graphs satisfy the self similarity property. As a consequence, it could be used to study the general multigraph conjecture for larger graph classes.
| 2022 | |
| Agencia Nacional de Investigación e Innovación | |
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Network Reliability Gross-Saccoman multigraph conjecture Graph Theory Uniformly Most Reliable Graph Self Similarity Property Multigraph Ciencias Naturales y Exactas Matemáticas Matemática Aplicada |
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| Inglés | |
| Agencia Nacional de Investigación e Innovación | |
| REDI | |
| https://hdl.handle.net/20.500.12381/701 | |
| Acceso abierto | |
| Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional. (CC BY-NC-ND) |
| Sumario: | An enigmatic conjecture in network synthesis asserts that the the uniformly most reliable multigraphs are simple. Daniel Gross and John Saccoman proved in 1998 that the answer is affirmative whenever m ≤ n + 2, where n and m is the respective number of nodes and edges of the multigraphs. They conjectured that the optimality is also achieved by simple graphs when m = n + 3. A proof for this conjecture recently appeared. In this article we provide a unified short proof for the previous cases where m ≤ n + 3. Our proof strategy holds whenever the most reliable simple graphs satisfy the self similarity property. As a consequence, it could be used to study the general multigraph conjecture for larger graph classes. |
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