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Circuit covers of cubic signed graphs.
- Source :
-
Journal of Graph Theory . Sep2018, Vol. 89 Issue 1, p40-54. 15p. - Publication Year :
- 2018
-
Abstract
- Abstract: A signed graph, denoted by ( G , σ ), is a graph G associated with a mapping σ : E ( G ) → { − 1 , + 1 }. A cycle of ( G , σ ) is a connected 2‐regular subgraph. A cycle C is positive if it has an even number of negative edges, and negative otherwise. A signed‐circuit of a signed graph ( G , σ ) is a positive cycle or a barbell consisting of two edge‐disjoint negative cycles joined by a path. The definition of a signed‐circuit of signed graph comes from the signed‐graphic matroid. A signed‐circuit cover of ( G , σ ) is a family of signed‐circuits covering all edges of ( G , σ ). A signed‐circuit cover with the smallest total length is called a shortest signed‐circuit cover of ( G , σ ) and its length is denoted by scc ( G , σ ). Bouchet proved that a signed graph has a signed‐circuit cover if and only if it is flow‐admissible (i.e., has a nowhere‐zero integer flow). In this article, we show that a 3‐connected flow‐admissible signed graph does not necessarily have a signed‐circuit double cover. For shortest signed‐circuit cover of 2‐edge‐connected cubic signed graphs ( G , σ ), we show that scc ( G , σ ) < 26 | E ( G ) | / 9 if it is flow‐admissible. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 03649024
- Volume :
- 89
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Journal of Graph Theory
- Publication Type :
- Academic Journal
- Accession number :
- 130628211
- Full Text :
- https://doi.org/10.1002/jgt.22238