Shortest circuit covers of signed graphs

Cheng, Jun; Lu, You; Luo, Rong; Zhang, Cun-Quan

doi:10.48550/arxiv.1510.05717

Cited by 3 publications

(9 citation statements)

References 20 publications

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“…Currently the best bound for this problem is due to Cheng et al [16]. Let the negativeness of a signed graph be the minimum number of negative edges in any equivalent signed graph.…”

Section: Circuit Coversmentioning

confidence: 99%

Nowhere-zero flows in signed graphs: A survey

Kaiser¹,

Rollová²,

Lukoťka³

2016

Preprint

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“…Currently the best bound for this problem is due to Cheng et al [16]. Let the negativeness of a signed graph be the minimum number of negative edges in any equivalent signed graph.…”

Section: Circuit Coversmentioning

confidence: 99%

Nowhere-zero flows in signed graphs: A survey

Kaiser¹,

Rollová²,

Lukoťka³

2016

Preprint

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“…Theorem 1.5 (Cheng, Lu, Luo and Zhang [5]). Let (G, σ) be a 2-edge-connected signed graph with even negativeness.…”

Section: Introductionmentioning

confidence: 99%

“…al. [5] for 2-edge-connected signed graph (G, σ) with even negativeness (see Section 2 for definition of negtiveness) as follows.…”

Section: Introductionmentioning

confidence: 99%

“…Then scc(G, σ) ≤ 11|E(G)|/3. For 2-connected cubic signed graphs (G, σ) with even negativeness, the above bound could be improved to scc(G, σ) ≤ 3|E(G)| + ǫ(G, σ)/3 in terms of negativeness ǫ(G, σ) of (G, σ) (see [5]). In this paper, we consider the shortest circuit cover of cubic signed graphs and the following is our main result.…”

Section: Introductionmentioning

confidence: 99%

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Circuit covers of cubic signed graphs

2018

Journal of Graph Theory

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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.

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“…Theorem 1.2 (Bermond, Jackson and Jaeger [2], Alon and Tarsi [1]). Every bridgeless graph G has a circuit cover with length at most 5 3 |E(G)|.…”

Section: Introductionmentioning

confidence: 99%

Minimum $T$-Joins and Signed-Circuit Covering

Wu¹,

Ye²

2018

Preprint

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Let G be a graph and T be a vertex subset of G with even cardinality. A T -join of G is a subset J of edges such that a vertex of G is incident with an odd number of edges in J if and only if the vertex belongs to T . Minimum T -joins have many applications in combinatorial optimizations. In this paper, we show that a minimum T -join of a connected graph G has at most |E(G)| − 1 2 |E( G )| edges where G is the maximum bidegeless subgraph of G. Further, we are able to use this result to show that every flow-admissible signed graph (G, σ) has a signed-circuit cover with length at most 19 6 |E(G)|. Particularly, a 2-edge-connected signed graph (G, σ) with even negativeness has a signed-circuit cover with length at most 8 3 |E(G)|.

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Shortest circuit covers of signed graphs

Cited by 3 publications

References 20 publications

Nowhere-zero flows in signed graphs: A survey

Nowhere-zero flows in signed graphs: A survey

Circuit covers of cubic signed graphs

Minimum $T$-Joins and Signed-Circuit Covering

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