Shortest coverings of graphs with cycles

Bermond, J.-C.; Jackson, Bill; Jaeger, François

doi:10.1016/0095-8956(83)90056-4

Cited by 101 publications

(102 citation statements)

References 8 publications

Supporting

Mentioning

100

Contrasting

Unclassified

Order By: Relevance

“…The following result of Alon and Tarsi [1] and Bermond, Jackson and Jaeger [2] is the best known general result on the length of cycle covers.…”

Section: Short Cycle Coversmentioning

confidence: 99%

1‐Factor and Cycle Covers of Cubic Graphs

Steffen

2014

Journal of Graph Theory

View full text Add to dashboard Cite

Let G be a bridgeless cubic graph. Consider a list of k 1-factors of G. Let E i be the set of edges contained in precisely i members of the k 1-factors. Let µ k (G) be the smallest |E 0 | over all lists of k 1-factors of G.Any list of three 1-factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Bergecovers and for the existence of three 1-factors with empty intersection. Furthermore, if µ 3 (G) = 0, then 2µ 3 (G) is an upper bound for the girth of G. We also prove some new upper bounds for the length of shortest cycle covers of bridgeless cubic graphs.Cubic graphs with µ 4 (G) = 0 have a 4-cycle cover of length 4 3 |E(G)| and a 5-cycle double cover. These graphs also satisfy two conjectures of Zhang [18]. We also give a negative answer to a problem stated in [18].

show abstract

“…The following result of Alon and Tarsi [1] and Bermond, Jackson and Jaeger [2] is the best known general result on the length of cycle covers.…”

Section: Short Cycle Coversmentioning

confidence: 99%

1‐Factor and Cycle Covers of Cubic Graphs

Steffen

2014

Journal of Graph Theory

View full text Add to dashboard Cite

show abstract

“…Bermond, Jackson, and Jaeger [1] showed that every 2-edge-connectedgraph has a collection of seven even subgraphs so that every edge is in exactly four of these subgraphs. This easily implies that every 2-edge-connected graph has an 7-dimensional unit distance flow.…”

Section: -Flowmentioning

confidence: 99%

Group flow, complex flow, unit vector flow, and the(2+ϵ)-flow conjecture

Thomassen¹

2014

Journal of Combinatorial Theory, Series B

View full text Add to dashboard Cite

“…We do so by bounding the summation through a product of terms that involves what we call apples-an apple is a simple cycle or a cycle plus a connected line. Along with the result of Bermond, Jackson, and Jaeger [3], this leads to the eventual result that the error in the Bethe approximation for the number of independent sets decays as Oðn −γ Þ for some γ > 0 for any graph on n nodes with max-degree d and girth larger than 8d log 2 n.…”

Section: Our Resultsmentioning

confidence: 99%

Counting Independent Sets Using the Bethe Approximation

Chandrasekaran¹,

Chertkov²,

Gamarnik³

et al. 2011

SIAM J. Discrete Math.

View full text Add to dashboard Cite

Abstract. We consider the #P-complete problem of counting the number of independent sets in a given graph. Our interest is in understanding the effectiveness of the popular belief propagation (BP) heuristic. BP is a simple iterative algorithm that is known to have at least one fixed point, where each fixed point corresponds to a stationary point of the Bethe free energy (introduced by Yedidia, Freeman, and Weiss [IEEE Trans. Inform. Theory, 51 (2004), pp. 2282-2312 in recognition of Bethe's earlier work in 1935). The evaluation of the Bethe free energy at such a stationary point (or BP fixed point) leads to the Bethe approximation for the number of independent sets of the given graph. BP is not known to converge in general, nor is an efficient, convergent procedure for finding stationary points of the Bethe free energy known. Furthermore, the effectiveness of the Bethe approximation is not well understood. As the first result of this paper we propose a BP-like algorithm that always converges to a stationary point of the Bethe free energy for any graph for the independent set problem. This procedure finds an ε-approximate stationary point in Oðn 2 d 4 2 d ε −4 log 3 ðnε −1 ÞÞ iterations for a graph of n nodes with max-degree d. We study the quality of the resulting Bethe approximation using the recently developed "loop series" framework of Chertkov and Chernyak [J. Stat. Mech. Theory Exp., 6 (2006), P06009]. As this characterization is applicable only for exact stationary points of the Bethe free energy, we provide a slightly modified characterization that holds for ε-approximate stationary points. We establish that for any graph on n nodes with max-degree d and girth larger than 8d log 2 n, the multiplicative error between the number of independent sets and the Bethe approximation decays as 1 þ Oðn −γ Þ for some γ > 0. This provides a deterministic counting algorithm that leads to strictly different results compared to a recent result of Weitz [in Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, ACM Press, New York, 2006, pp. 140-149]. Finally, as a consequence of our analysis we prove that the Bethe approximation is exceedingly good for a random 3-regular graph conditioned on the shortest cycle cover conjecture of Alon and Tarsi [SIAM J. Algebr. Discrete Methods, 6 (1985), pp. 345-350] being true.

show abstract

Shortest coverings of graphs with cycles

Cited by 101 publications

References 8 publications

1‐Factor and Cycle Covers of Cubic Graphs

1‐Factor and Cycle Covers of Cubic Graphs

Group flow, complex flow, unit vector flow, and the(2+ϵ)-flow conjecture

Counting Independent Sets Using the Bethe Approximation

Contact Info

Product

Resources

About