Nowhere-Zero Flows on Signed Wheels and Signed Fans

Hu, Lili; Li, Xiangwen

doi:10.1007/s40840-016-0416-1

Cited by 5 publications

(3 citation statements)

References 15 publications

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“…Hu and Li [7] show that

({W}_{5},{\sigma }^{* })

in Figure 1 admits a 5‐NZF but no 4‐NZF. Then

(G,\sigma )

does not admit a 4‐NZF.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

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Integer flows on triangularly connected signed graphs

Li,

Luo

et al. 2024

Journal of Graph Theory

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A triangle‐path in a graph is a sequence of distinct triangles in such that for any with , and if . A connected graph is triangularly connected if for any two nonparallel edges and there is a triangle‐path such that and . For ordinary graphs, Fan et al. characterize all triangularly connected graphs that admit nowhere‐zero 3‐flows or 4‐flows. Corollaries of this result include the integer flow of some families of ordinary graphs, such as locally connected graphs due to Lai and some types of products of graphs due to Imrich et al. In this paper, Fan's result for triangularly connected graphs is further extended to signed graphs. We proved that a flow‐admissible triangularly connected signed graph admits a nowhere‐zero 4‐flow if and only if it is not the wheel associated with a specific signature. Moreover, this result is sharp since there are infinitely many unbalanced triangularly connected signed graphs admitting a nowhere‐zero 4‐flow but no 3‐flow.

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“…Hu and Li [7] show that

({W}_{5},{\sigma }^{* })

in Figure 1 admits a 5‐NZF but no 4‐NZF. Then

(G,\sigma )

does not admit a 4‐NZF.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…Integer flows on signed graphs also have been studied for many specific families of graphs. For more details, the readers are referred to [3, 7, 9, 11, 13–17, 20, 22].…”

Section: Introductionmentioning

confidence: 99%

Integer flows on triangularly connected signed graphs

Li,

Luo

et al. 2024

Journal of Graph Theory

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show abstract

“…Theorem 3.4 doesn't hold for = 4. There is a signed W 5 which has a nowhere-zero Z 4 -flow but doesn't have a nowhere-zero 4-flow (see [3]).…”

Section: Circuit Decomposition and Sum Of 2-flowsmentioning

confidence: 99%

Flows on signed graphs without long barbells

Luo

Schubert

et al. 2019

Preprint

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Many basic properties in Tutte's flow theory for unsigned graphs do not have their counterparts for signed graphs. However, signed graphs without long barbells in many ways behave like unsigned graphs from the point view of flows. In this paper, we study whether some basic properties in Tutte's flow theory remain valid for this family of signed graphs. Specifically let (G, σ) be a flow-admissible signed graph without long barbells. We show that it admits a nowhere-zero 6-flow and that it admits a nowhere-zero modulo k-flow if and only if it admits a nowhere-zero integer k-flow for each integer k ≥ 3 and k = 4. We also show that each nowhere-zero positive integer k-flow of (G, σ) can be expressed as the sum of some 2-flows. For general graphs, we show that every nowhere-zero p q -flow can be normalized in such a way, that each flow value is a multiple of 1 2q . As a consequence we prove the equality of the integer flow number and the ceiling of the circular flow number for flow-admissible signed graphs without long barbells.

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Flow number of signed Halin graphs

Wang

Zhang

2021

Applied Mathematics and Computation

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Nowhere-Zero Flows on Signed Wheels and Signed Fans

Cited by 5 publications

References 15 publications

Integer flows on triangularly connected signed graphs

Integer flows on triangularly connected signed graphs

Flows on signed graphs without long barbells

Flow number of signed Halin graphs

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