Additive bases and flows in graphs

Verclos, Rémi de Joannis de; Thomassé, Stéphan

doi:10.1016/j.endm.2017.06.066

Cited by 4 publications

(12 citation statements)

References 18 publications

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“…One example is any Z 5 -flow that uses only values 1 and 2. Esperet, de Verclos, Le, and Thomassé [6] proved that if a graph G is strongly Z 5 -connected, then every orientation D(G) of G admits an antisymmetric Z 5 -flow. Together with work of Lovász et al [13], this implies that every directed 12-edge-connected graph admits an antisymmetric Z 5 -flow.…”

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confidence: 99%

Circular Flows in Planar Graphs

Cranston¹,

Li²

2020

SIAM J. Discrete Math.

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For integers a ≥ 2b > 0, a circular a/b-flow is a flow that takes values from {±b, ±(b+1), . . . , ±(a−b)}. The Planar Circular Flow Conjecture states that every 2k-edge-connected planar graph admits a circular (2 + 2 k )-flow. The cases k = 1 and k = 2 are equivalent to the Four Color Theorem and Grötzsch's 3-Color Theorem. For k ≥ 3, the conjecture remains open. Here we make progress when k = 4 and k = 6. We prove that (i) every 10-edge-connected planar graph admits a circular 5/2-flow and (ii) every 16-edge-connected planar graph admits a circular 7/3-flow. The dual version of statement (i) on circular coloring was previously proved by Dvořák and Postle (Combinatorica 2017), but our proof has the advantages of being much shorter and avoiding the use of computers for case-checking. Further, it has new implications for antisymmetric flows. Statement (ii) is especially interesting because the counterexamples to Jaeger's original Circular Flow Conjecture are 12-edge-connected nonplanar graphs that admit no circular 7/3-flow. Thus, the planarity hypothesis of (ii) is essential. 1 Jaeger [9] showed that if p, q, r, s ∈ Z + and p/q = r/s, then each graph G has a circular p/q-flow if and only if it has a circular r/s-flow. (See [7] for more details.) We use this result implicitly in the present paper.

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confidence: 99%

Circular Flows in Planar Graphs

Cranston¹,

Li²

2020

SIAM J. Discrete Math.

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“…The girth condition is reduced to 14 in [11], to 13 in [10], and finally to 12 in [9]. By duality, using the results of Nešetřil and Raspaud [69], the result of Esperet et al [28] together with the theorem of Lovász et al [59] also showed girth 12 suffices. Esperet et al [28] remarked that "it is not known whether the same holds for planar graphs of girth at least 11 after the grith 12 result of Borodin, Ivanova and Kostochka [9] since 2007".…”

Section: Introductionmentioning

confidence: 93%

“…By duality, using the results of Nešetřil and Raspaud [69], the result of Esperet et al [28] together with the theorem of Lovász et al [59] also showed girth 12 suffices. Esperet et al [28] remarked that "it is not known whether the same holds for planar graphs of girth at least 11 after the grith 12 result of Borodin, Ivanova and Kostochka [9] since 2007". Note that, the result of Dvořák and Postle [26] seems not to apply to homomorphism to oriented graphs.…”

Section: Introductionmentioning

confidence: 94%

“…Note that, the result of Dvořák and Postle [26] seems not to apply to homomorphism to oriented graphs. In this chapter, we improve both bounds on the problems of Lai [49] and Esperet et al [28] to 11 for planar graph. Again, by duality on Theorems 6.1.2 and 6.1.3 (see [30] and [69] ), we have the following.…”

Section: Introductionmentioning

confidence: 97%

“…Esperet, De Verclos, Le and Thomassé [28] proved that if a graph G is strongly Z 5connected, then for any orientation D of G, the directed graph D(G) admits an antisymmetric Z 5 -flow. Thus exploring the results of Lovász et al [59], every directed 12-edgeconnected graph admits an antisymmetric Z 5 -flow, and they conjectured [28] that every directed 8-edge-connected graph admits an antisymmetric Z 5 -flow. Here, an antisymmetric Z 5 -flow in a directed graph D = D(G) is a Z 5 -flow (D, f ) such that f (e) = −f (e ) for any pair of distinct edges e, e ∈ E(G).…”

Section: Introductionmentioning

confidence: 99%

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Group Connectivity and Modulo Orientations of Graphs

Li¹

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This dissertation focuses on the subject of nowhere-zero flow problems on graphs. Tutte's 5-Flow Conjecture (1954) states that every bridgeless graph admits a nowhere-zero 5-flow, and Tutte's 3-Flow Conjecture (1972) states that every 4-edge-connected graph admits a nowherezero 3-flow. Jaeger et al. (1992) further conjectured that every 5-edge-connected graph is Z 3-connected, whose truth implies the 3-Flow Conjecture. Extending Tutte's flows conjectures, Jaeger's Circular Flow Conjecture (1981) says every 4p-edge-connected graph admits a modulo (2p + 1)-orientation, that is, an orientation such that the indegree is congruent to outdegree modulo 2p + 1 at every vertex. Note that the p = 1 case of Circular Flow Conjecture coincides with the 3-Flow Conjecture, and the case of p = 2 implies the 5-Flow Conjecture. This work is devoted to provide some partial results on these problems. It is proved in Chapter 2 that every graph with four edge-disjoint spanning trees is Z 3connected. Consequently, Jaeger et al.'s group connectivity conjecture and Tutte's 3-Flow Conjecture hold for 5-edge-connected essentially 23-edge-connected graphs. We also provide several equivalent versions of Jaeger et al.'s group connectivity conjecture and indicate that it is enough to verify the conjecture for 5-edge-connected essentially 8-edge-connected graphs. In Chapter 3, Tutte's 3-Flow Conjecture is verified for graphs with independence number at most 4. The relation of orientation and group connectivity is studied in Chapter 4. It shows that every strongly Z m-connected graph contains m − 1 edge-disjoint spanning trees, and hence every Z m-connected graph G has (m − 1)(|V (G)| − 1)/(m − 2) edges, which solves a conjecture of Luo et al. (2012). Those results are applied to establish some monotonicity properties of group connectivity that every strongly Z 5-connected graph is Z 3-connected, and every Z 3-connected graph is A-connected for any Abelian group A with size |A| ≥ 4. Infinite families of counterexamples to Jaeger's Circular Flow Conjecture are presented in Chapter 5. For p ≥ 3, there are 4p-edge-connected graphs not admitting modulo (2p + 1)orientation; for p ≥ 5, there are (4p + 1)-edge-connected graphs not admitting modulo (2p + 1)orientation. Towards the p = 2 case of Circular Flow Conjecture and the 5-Flow Conjecture, we show in Chapter 6 that every 10-edge-connected planar graph admits a modulo 5-orientation.

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Weighted Modulo Orientations of Graphs and Signed Graphs

Han

Lai

2022

Electron. J. Combin.

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Given a graph $G$ and an odd prime $p$, for a mapping $f: E(G) \to {\mathbb Z}_p\setminus\{0\}$ and a ${\mathbb Z}_p$-boundary $b$ of $G$, an orientation $\tau$ is called an $(f,b;p)$-orientation if the net out $f$-flow is the same as $b(v)$ in ${\mathbb Z}_p$ at each vertex $v\in V(G)$ under orientation $D$. This concept was introduced by Esperet et al. (2018), generalizing mod $p$-orientations and closely related to Tutte's nowhere zero 3-flow conjecture. They proved that $(6p^2 - 14p + 8)$-edge-connected graphs have all possible $(f,b;p)$-orientations. In this paper, the framework of such orientations is extended to signed graph through additive bases. We also study the $(f,b;p)$-orientation problem for some (signed) graphs families including complete graphs, chordal graphs, series-parallel graphs and bipartite graphs, indicating that much lower edge-connectivity bound still guarantees the existence of such orientations for those graph families.

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Additive bases and flows in graphs

Abstract: International audienc

Cited by 4 publications

References 18 publications

Circular Flows in Planar Graphs

Circular Flows in Planar Graphs

Group Connectivity and Modulo Orientations of Graphs

Weighted Modulo Orientations of Graphs and Signed Graphs

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