Claw‐decompositions and tutte‐orientations

Barát, János; Thomassen, Carsten

doi:10.1002/jgt.20149

Cited by 58 publications

(50 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As seen in Proposition 1.2, "Z 3 -flow contractible" is equivalent to "Z 3 -connected," which is the same as "admitting all generalized Tutte-orientations" defined by Barát and Thomassen [1]. Thus, Theorem 4.5 in [1] can be restated as:…”

Section: Introductionmentioning

confidence: 71%

Nowhere-zero 3-flows in triangularly connected graphs

Fan

Lai

et al. 2008

Journal of Combinatorial Theory, Series B

View full text Add to dashboard Cite

Let H 1 and H 2 be two subgraphs of a graph G. We say that G is the 2-sum of H 1 and H 2 , denoted byA connected graph G is triangularly connected if for any two edges e and e , which are not parallel, there is a triangle-path T 1 T 2 · · · T m such that e ∈ E(T 1 ) and e ∈ E(T m ). Let G be a triangularly connected graph with at least three vertices. We prove that G has no nowhere-zero 3-flow if and only if there is an odd wheel W and a subgraph G 1 such that G = W ⊕ 2 G 1 , where G 1 is a triangularly connected graph without nowhere-zero 3-flow. Repeatedly applying the result, we have a complete characterization of triangularly connected graphs which have no nowhere-zero 3-flow. As a consequence, G has a nowhere-zero 3-flow if it contains at most three 3-cuts. This verifies Tutte's 3-flow conjecture and an equivalent version by Kochol for triangularly connected graphs. By the characterization, we obtain extensions to earlier results on locally connected graphs, chordal graphs and squares of graphs. As a corollary, we obtain a result of Barát and Thomassen that every triangulation of a surface admits all generalized Tutte-orientations.

show abstract

Section: Introductionmentioning

confidence: 71%

Nowhere-zero 3-flows in triangularly connected graphs

Fan

Lai

et al. 2008

Journal of Combinatorial Theory, Series B

View full text Add to dashboard Cite

show abstract

“…Note these two theorems are also generalizations of Theorem 3.6. [1].) Every 4 log 2 n -edge-connected graph with n vertices is Z 3 -connected.…”

Section: Lemma 55 (See Seymourmentioning

confidence: 98%

“…Every 4 log 2 n -edge-connected graph with n vertices is Z 3 -connected. [1].) Let G be a graph with n vertices.…”

Section: Lemma 55 (See Seymourmentioning

confidence: 99%

Flows and parity subgraphs of graphs with large odd-edge-connectivity

Shu

Zhang

2012

Journal of Combinatorial Theory, Series B

View full text Add to dashboard Cite

The odd-edge-connectivity of a graph G is the size of the smallest odd edge cut of G. Tutte conjectured that every odd-5-edgeconnected graph admits a nowhere-zero 3-flow. As a weak version of this famous conjecture, Jaeger conjectured that there is an integer k such that every k-edge-connected graph admits a nowhere-zero 3-flow. Jaeger [F. Jaeger, Flows and generalized coloring theorems in graphs, J. Combin. Theory Ser. B 26 (1979) 205-216] proved that every 4-edge-connected graph admits a nowhere-zero 4-flow. Galluccio and Goddyn [A. Galluccio, L.A. Goddyn, The circular flow number of a 6-edge-connected graph is less than four, Combinatorica 22 (2002) 455-459] proved that the flow index of every 6-edge-connected graph is strictly less than 4. This result is further strengthened in this paper that the flow index of every odd-7-edge-connected graph is strictly less than 4. The second main result in this paper solves an open problem that every odd-(2k + 1)-edge-connected graph contains k edge-disjoint parity subgraphs. The third main theorem of this paper proves that if the odd-edge-connectivity of a graph G is at least 4 log 2 |V (G)| + 1, then G admits a nowhere-zero 3-flow. This result is a partial result to the weak 3-flow conjecture by Jaeger and improves an earlier result by Lai et al. The fourth main result of this paper proves that every odd-(4t + 1)-edge-connected graph G has a circular (2t + 1) even subgraph double cover. This result generalizes an earlier result of Jaeger.

show abstract

“…Barát and Thomassen [1] considered whether there is a particular edge-connectivity k c so that every k c -edge-connected graph G with |E(G)| a multiple of 3 has a claw-decomposition (for a simple graph, this means its edge set partitions into sets of size 3, each inducing a K 1,3 ). Thomassen [13] recently showed edge-connectivity 8 suffices.…”

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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Claw‐decompositions and tutte‐orientations

Cited by 58 publications

References 16 publications

Nowhere-zero 3-flows in triangularly connected graphs

Nowhere-zero 3-flows in triangularly connected graphs

Flows and parity subgraphs of graphs with large odd-edge-connectivity

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

Contact Info

Product

Resources

About