Mod (2p + 1)-Orientations and $K_{1,2p+1}$-Decompositions

Lai, Hong‐Jian

doi:10.1137/060676945

Cited by 32 publications

(36 citation statements)

References 9 publications

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“…By modifying a famous Jaeger's conjecture on a mod

k

‐orientation, Lai conjectured in 2007 that for an odd integer

k

with

k \geq 3

, every (

2 k - 2

)‐edge‐connected graph is strongly

Z_{k}

‐connected. Recently, this conjecture (and even Jaeger's one) was disproved by Han, Li, We, and Zhang , and they posed a new conjecture on a mod

k

‐orientation.…”

Section: Conjecture 1 and Strongly Zk‐connectivitymentioning

confidence: 99%

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Orientations of graphs avoiding given lists on out‐degrees

Akbari

Dalirrooyfard

Ehsani

et al. 2019

Journal of Graph Theory

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Let G be a graph and F : V ( G ) → 2 N be a mapping. The graph G is said to be F‐ avoiding if there exists an orientation O of G such that d O + ( v ) ∉ F ( v ) for every v ∈ V ( G ), where d O + ( v ) denotes the out‐degree of v in the directed graph G with respect to O. In this paper it is shown that if G is bipartite and ∣ F ( v ) ∣ ≤ d G ( v ) / 2 for every v ∈ V ( G ), then G is F‐avoiding. The bound ∣ F ( v ) ∣ ≤ d G ( v ) / 2 is best possible. For every graph G, we conjecture that if ∣ F ( v ) ∣ ≤ 1 2 ( d G ( v ) − 1 ) for every v ∈ V ( G ), then G is F‐avoiding. We also argue about this conjecture for the best possibility of the conditions and also show some partial solutions.

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“…By modifying a famous Jaeger's conjecture on a mod

k

‐orientation, Lai conjectured in 2007 that for an odd integer

k

with

k \geq 3

, every (

2 k - 2

)‐edge‐connected graph is strongly

Z_{k}

‐connected. Recently, this conjecture (and even Jaeger's one) was disproved by Han, Li, We, and Zhang , and they posed a new conjecture on a mod

k

‐orientation.…”

Section: Conjecture 1 and Strongly Zk‐connectivitymentioning

confidence: 99%

“…572–573). Note that such an

F

‐avoiding orientation is called a mod

3

‐orientation, see and Section in this paper.…”

Section: Introductionmentioning

confidence: 99%

Orientations of graphs avoiding given lists on out‐degrees

Akbari

Dalirrooyfard

Ehsani

et al. 2019

Journal of Graph Theory

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“…It was conjectured by Lai [13] that for every k 1, every (4k + 1)edge-connected graph G has a β-orientation for every Z 2k+1 -boundary β of G. If true, this conjecture would directly imply (using the same proof as that of Theorem 12) that for any k 2, every directed ⌈ 4k+1 k−1 ⌉edge-connected graph has a Z 2k+1 -asf. In particular, this would show that directed 5-edge-connected graph have a Z 13 -asf, directed 6-edgeconnected graph have a Z 9 -asf, directed 7-edge-connected graph have a Z 7 -asf, and directed 9-edge-connected graph have a Z 5 -asf.…”

Section: Orientations and Flows In Graphsmentioning

confidence: 94%

Additive bases and flows in graphs

Verclos¹,

Thomassé²

2017

Electronic Notes in Discrete Mathematics

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“…In particular, a 5-edge-connected planar graph G has a claw-decomposition if |E(G)| a multiple of 3. Lai [7] presented a 4-edge-connected example that shows that 5-edge-connected cannot be reduced to 4-edge-connected.…”

Section: Introductionmentioning

confidence: 99%

Group-colouring, group-connectivity, claw-decompositions, and orientations in 5-edge-connected planar graphs

Richter

Thomassen

Younger

2016

Journal of Combinatorics

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Let G be a graph, let Γ be an Abelian group with identity 0 Γ , and, for each vertex v of G, let p(v) be a prescription such that v∈V (G) p(v) = 0 Γ . A (Γ, p)-flow consists of an orientation D of G and, for each edge e of G, a label f (e) in Γ \ {0 Γ } such that, for each vertex v of G,If such an orientation D and labelling f exists for all such p, then G is Γ-connected .Our main result is that if G is a 5-edge-connected planar graph and |Γ| ≥ 3, then G is Γ-connected. This is equivalent to a dual colourability statement proved by Lai and Li (2007): planar graphs with girth at least 5 are "Γ-colourable". Our proof is considerably shorter than theirs. Moreover, the Γ-colourability result of Lai and Li is already a consequence of Thomassen's (2003) 3-list-colour proof for planar graphs of girth at least 5.Our theorem (as well as the girth 5 colourability result) easily implies that every 5-edge-connected planar graph for which |E(G)| is a multiple of 3 has a claw decomposition, resolving a question of Barát and Thomassen. It also easily implies the dual of Grötzsch's Theorem, that every planar graph without 1-or 3-cut has a 3-flow; this is equivalent to Grötzsch's Theorem.

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Mod (2p + 1)-Orientations and $K_{1,2p+1}$-Decompositions

Cited by 32 publications

References 9 publications

Orientations of graphs avoiding given lists on out‐degrees

Orientations of graphs avoiding given lists on out‐degrees

Additive bases and flows in graphs

Group-colouring, group-connectivity, claw-decompositions, and orientations in 5-edge-connected planar graphs

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