Cuts in matchings of 3-connected cubic graphs

Knauer, Kolja; Valicov, Petru

doi:10.1016/j.ejc.2018.09.004

Cited by 6 publications

(4 citation statements)

References 14 publications

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“…The conjecture is part of an active field of research. It has been verified for planar oriented graphs on at most 26 vertices [20] and holds for planar digraphs of digirth at least 4 [25].…”

Section: Kronk and Whitementioning

confidence: 91%

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On the Dichromatic Number of Surfaces

Aboulker¹,

Havet²,

Knauer³

et al. 2022

Electron. J. Combin.

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In this paper, we give bounds on the dichromatic number $\vec{\chi}(\Sigma)$ of a surface $\Sigma$, which is the maximum dichromatic number of an oriented graph embeddable on $\Sigma$. We determine the asymptotic behaviour of $\vec{\chi}(\Sigma)$ by showing that there exist constants $a_1$ and $a_2$ such that, $ a_1\frac{\sqrt{-c}}{\log(-c)} \leq \vec{\chi}(\Sigma) \leq a_2 \frac{\sqrt{-c}}{\log(-c)} $ for every surface $\Sigma$ with Euler characteristic $c\leq -2$. We then give more explicit bounds for some surfaces with high Euler characteristic. In particular, we show that the dichromatic numbers of the projective plane $\mathbb{N}_1$, the Klein bottle $\mathbb{N}_2$, the torus $\mathbb{S}_1$, and Dyck's surface $\mathbb{N}_3$ are all equal to $3$, and that the dichromatic numbers of the $5$-torus $\mathbb{S}_5$ and the $10$-cross surface $\mathbb{N}_{10}$ are equal to $4$. We also consider the complexity of deciding whether a given digraph or oriented graph embeddable on a fixed surface is $k$-dicolourable. In particular, we show that for any fixed surface, deciding whether a digraph embeddable on this surface is $2$-dicolourable is NP-complete, and that deciding whether a planar oriented graph is $2$-dicolourable is NP-complete unless all planar oriented graphs are $2$-dicolourable (which was conjectured by Neumann-Lara).

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Section: Kronk and Whitementioning

confidence: 91%

“…20. But H is not 2-dicolourable, so it contains a 3-dicritical oriented subgraph H, and m( H) 20. By Proposition 18, there is a unique such 3-dicritical oriented graph and it has 7 vertices and 20 arcs.…”

Section: Projective Plane Torus Klein Bottle and Dyck's Surfacementioning

confidence: 98%

On the Dichromatic Number of Surfaces

Aboulker¹,

Havet²,

Knauer³

et al. 2022

Electron. J. Combin.

Self Cite

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“…In [16] it is written that Lomonosov and Kelmans proved without computer assistance that Tutte's conjecture holds up to 30 vertices, but no reference is given. Recently, Knauer and Valicov [18] verified Tutte's conjecture up to 40 vertices using computational methods. Our aim in this paper is to prove that the Georges-Kelmans graph is in fact minimal.…”

Section: Introductionmentioning

confidence: 99%

The Minimality of the Georges-Kelmans Graph

Brinkmann,

Goedgebeur,

McKay

2021

Preprint

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In 1971, Tutte wrote in an article that it is tempting to conjecture that every 3connected bipartite cubic graph is hamiltonian. Motivated by this remark, Horton constructed a counterexample on 96 vertices. In a sequence of articles by different authors several smaller counterexamples were presented. The smallest of these graphs is a graph on 50 vertices which was discovered independently by Georges and Kelmans. In this article we show that there is no smaller counterexample. As all non-hamiltonian 3-connected bipartite cubic graphs in the literature have cyclic 4-cuts-even if they have girth 6-it is natural to ask whether this is a necessary prerequisite. In this article we answer this question in the negative and give a construction of an infinite family of non-hamiltonian cyclically 5-connected bipartite cubic graphs.In 1969 Barnette gave a weaker version of the conjecture stating that 3-connected planar bipartite cubic graphs are hamiltonian. We show that Barnette's conjecture is true up to at least 90 vertices. We also report that a search of small non-hamiltonian 3-connected bipartite cubic graphs did not find any with genus less than 4.

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“…A graph is k-regular if the degree of each vertex is equal to k. In general, 3-regular graphs are also called cubic graphs, which play a prominent role in graph theory. During recent years, many investigations concerning cubic graphs were reported (see, e.g., in [25,27,28,6,26,5]).…”

Section: Introductionmentioning

confidence: 99%

Cubic bipartite graphs with minimum spectral gap

Liu¹,

Xue²

2022

Preprint

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Algebraic connectivity of a graph is the second smallest eigenvalue of its Laplacian matrix. Abdi, Ghorbani and Imrich, in [European J. Combin. 95 (2021) 103328], showed that the minimum algebraic connectivity of cubic graphs on 2n vertices is (1 + o(1)) π 2 2n 2 . The minimum value is attained on non-bipartite graphs. In this paper, we investigate the algebraic connectivity of cubic bipartite graphs. We prove that the minimum algebraic connectivity of cubic bipartite graphs on 2n vertices is (1 + o(1)) π 2 n 2 . Moreover, the unique cubic bipartite graph with minimum algebraic connectivity is determined. Based on the relation between the algebraic connectivity and spectral gap of regular graphs, the cubic bipartite graph with minimum spectral gap and the corresponding asymptotic value are also presented. In [J. Graph Theory 99 (2022) 671-690], Horak and Kim established a sharp upper bound for the number of perfect matchings in terms of the Fibonacci number. We obtain a spectral characterization for the extremal graphs by showing that a cubic bipartite graph has the maximum number of perfect matchings if and only if it minimizes the algebraic connectivity.

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Cuts in matchings of 3-connected cubic graphs

Cited by 6 publications

References 14 publications

On the Dichromatic Number of Surfaces

On the Dichromatic Number of Surfaces

The Minimality of the Georges-Kelmans Graph

Cubic bipartite graphs with minimum spectral gap

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