Construction of permutation snarks

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.

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‐edge colourable and several authors (see, eg, ) refer to a member of this subclass as a permutation snark.…”

Section: Ban‐linial's Conjecturementioning

confidence: 99%

Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

Abreu

Goedgebeur

Labbate

et al. 2019

“…F I G U R E 1 An example: The Petersen graph and a (8) -linked 2-factor { 1 , 2 } this direction is the verification ( [14,16]) of Fan-Raspaud conjecture for oddness two graphs, where Fan-Raspaud conjecture [5] is a weaker version of the Berge-Fulkerson conjecture. And, in [2], the conjecture was verified for some snarks of small orders.…”

mentioning

confidence: 99%

“…Let be a cubic graph and = { 1 , … , } be a 2-factor of such that | | is odd if and only if ≤ 2 for some integer ≤ 2 . The 2-factor is (8) -linked if, for every ≤ , there is a circuit of length 8 with edge sequence 1 … 8 where 1 , 5 ∈ ( 2 −1 ) and 3 , 7 ∈ ( 2 ). The circuit is called a (8) -link between 2 −1 and 2 .…”

mentioning

confidence: 99%

“…The 2-factor is (8) -linked if, for every ≤ , there is a circuit of length 8 with edge sequence 1 … 8 where 1 , 5 ∈ ( 2 −1 ) and 3 , 7 ∈ ( 2 ). The circuit is called a (8) -link between 2 −1 and 2 . And the cubic graph is (8) -linked if it contains a (8) -linked 2-factor.…”

mentioning

confidence: 99%

“…The circuit is called a (8) -link between 2 −1 and 2 . And the cubic graph is (8) -linked if it contains a (8) -linked 2-factor.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Berge–Fulkerson coloring for C₍₈₎‐linked graphs

Hao

Zhang

Zheng

2017

It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. Let G be a cubic graph and F={C1,…,Cr} be a 2‐factor of G such that false|Cjfalse| is odd if and only if j≤2k for some integer k. The 2‐factor F is C(8)‐linked if, for every i≤k, there is a circuit Di of length 8 with edge sequence e1i…e8i where e1i,e5i∈Efalse(C2i−1false) and e3i,e7i∈Efalse(C2ifalse). And the cubic graph G is C(8)‐linked if it contains a C(8)‐linked 2‐factor. It is proved in this article that every C(8)‐linked cubic graph is Berge–Fulkerson colorable. It is also noticed that many classical families of snarks (including some high oddness snarks) are C(8)‐linked. Consequently, the Berge–Fulkerson conjecture is verified for these infinite families of snarks.

Berge–Fulkerson coloring for C(12)‐linked permutation graphs

Liu

Hao

Zhang

et al. 2021

It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. Let G be a permutation graph with a 2‐factor MJX-tex-caligraphicscriptF = { C 1 , C 2 }. A circuit C 0 is MJX-tex-caligraphicscriptF‐alternating if E ( C 0 ) ⧹ ( E ( C 1 ) ∪ E ( C 2 ) ) is a perfect matching of C 0. A permutation graph G with a 2‐factor MJX-tex-caligraphicscriptF = { C 1 , C 2 } is C ( 12 )‐linked if it contains an MJX-tex-caligraphicscriptF‐alternating circuit of length at most 12. It is proved in this paper that every C ( 12 )‐linked permutation graph is Berge–Fulkerson colorable. As an application, the conjecture is verified for some families of snarks constructed by Abreu et al., Brinkmann et al., and Hägglund et al.