The structure of graphs with circular flow number 5 or more, and the complexity of their recognition problem

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A k‐bisection of a bridgeless cubic graph G is a 2‐colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes ( monochromatic components in what follows) have order at most k. Ban and Linial Conjectured that every bridgeless cubic graph admits a 2‐bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph G with ∣ E ( G ) ∣ ≡ 0 ( mod 2 ) has a 2 ‐edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (ie, a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we provide evidence of a strong relation of the conjectures of Ban‐Linial and Wormald with Ando's Conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above‐mentioned conjectures. Moreover, we prove Ban‐Linial's Conjecture for cubic‐cycle permutation graphs. As a by‐product of studying 2‐edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.

“…A stronger version of Theorem is proved in by Esperet, Tarsi and the fourth author (in terms of

(k + 2)

‐weak bisections). In particular, they prove the following:…”

Section: Ban‐linial's Conjecturementioning

confidence: 88%

Section: Ban-linial's Conjecturementioning

confidence: 93%

“…Theorem 2.5 (Esperet et al [17]). If a bridgeless cubic graph G does not admit a 2-bisection, then it has circular flow number at least 5.…”

Section: Ban-linial's Conjecturementioning

confidence: 99%

Section: Ban-linial's Conjecturementioning

confidence: 99%

2

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

Abreu

Goedgebeur

Labbate

et al. 2019

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Flows and Bisections in Cubic Graphs

Mazzuoccolo

Tarsi

2017

A k-weak bisection of a cubic graph G is a partition of the vertex-set of G into two parts V 1 and V 2 of equal size, such that each connected component of the subgraph of G induced by V i (i = 1, 2) is a tree of at most k − 2 vertices. This notion can be viewed as a relaxed version of nowhere-zero flows, as it directly follows from old results of Jaeger that every cubic graph G with a circular nowhere-zero r-flow has a r -weak bisection. In this paper we study problems related to the existence of k-weak bisections. We believe that every cubic graph which has a perfect matching, other than the Petersen graph, admits a 4-weak bisection and we present a family of cubic graphs with no perfect matching which do not admit such a bisection. The main result of this article is that every cubic graph admits a 5-weak bisection. When restricted to bridgeless graphs, that result would be a consequence of the assertion of the 5-flow Conjecture and as such it can be considered a (very small) step toward proving that assertion. However, the harder part of our proof focuses on graphs which do contain bridges.

Bounded‐excess flows in cubic graphs

Tarsi

2020

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An (r, α)-bounded excess flow ((r, α)-flow) in an orientation of a graph G = (V, E) is an assignment f : E → [1, r − 1], such that for every vertex x ∈ V , | e∈E + (x) f (e) − e∈E − (x) f (e)| ≤ α. E + (x), respectively E − (x), are the sets of edges directed from, respectively toward x. Bounded excess flows suggest a generalization of Circular nowhere zero flows, which can be regarded as (r, 0)-flows. We define (r, α) as Stronger or equivalent to (s, β) If the existence of an (r, α)-flow in a cubic graph always implies the existence of an (s, β)-flow in the same graph. We then study the structure of the bounded excess flow strength poset. Among other results, we define the Trace of a point in the r-α plane by tr(r, α) = r−2α 1−α and prove that among points with the same trace the stronger is the one with the smaller α (and larger r). e.g. If a cubic graph admits a k-nzf (trace k with α = 0) then it admits an (r, k−r k−2 )-flow for every r, 2 ≤ r ≤ k. A significant part of the article is devoted to proving the main result: Every cubic graph admits a (3 1 2 , 1 2 )-flow, and there exists a graph which does not admit any stronger bounded excess flow. Notice that tr(3 1 2 , 1 2 ) = 5 so it can be considered a step in the direction of the 5-flow Conjecture. Our result is the best possible for all cubic graphs while the seemingly stronger 5-flow Conjecture relates only to bridgeless graphs. We also show that if the circular flow number of a cubic graph is strictly less * Research supported in part by FWF-grant P27615-N25, headed by Herbert Fleischner than 5 then it admits a (3 1 3 , 1 3 )-flow (trace 4). We conjecture such a flow to exist in every cubic graph with a perfect matching, other than the Petersen graph. This conjecture is a stronger version of the Ban-Linial Conjecture [1], [4]. Our work here strongly rely on the notion of Orientable k-weak bisections, a certain type of k-weak bisections. k-weak bisections are defined and studied in [4].