Short cycle covers of graphs and nowhere-zero flows

Máčajová, Edita; Raspaud, André; Tarsi, Michael; Zhu, Xuding

doi:10.1002/jgt.20563

Cited by 10 publications

(10 citation statements)

References 15 publications

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“…This result is improved to smaller than 14 9 |E(G)| in [11]. We additionally deduce that there is an even 3-cycle cover of length smaller than 14 9 |E(G)|.…”

Section: Theorem 219mentioning

confidence: 57%

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1‐Factor and Cycle Covers of Cubic Graphs

Steffen

2014

Journal of Graph Theory

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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].

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“…This result is improved to smaller than 14 9 |E(G)| in [11]. We additionally deduce that there is an even 3-cycle cover of length smaller than 14 9 |E(G)|.…”

Section: Theorem 219mentioning

confidence: 57%

“…In [4] it is proved that if a cubic graph G has a Fulkerson-cover, then it has a 3-cycle cover of length at most 22 15 |E(G)|. This bound is best possible for 3-cycle covers of bridgeless cubic graphs, since it is attained by the Petersen graph [11]. We additionally show that there exists such a cycle cover which is even.…”

Section: Theorem 219mentioning

confidence: 84%

1‐Factor and Cycle Covers of Cubic Graphs

Steffen

2014

Journal of Graph Theory

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“…Jamshy, Raspaud, and Tarsi [7] showed that graphs with nowhere-zero 5-flow, which, according the Tutte's 5-flow conjecture, are all bridgeless graphs, have a cycle cover of length at most 1.6m. Máčajová et al [11] showed that the existence of a Fano-flow using at most 5 lines of the Fano plane on bridgeless cubic graphs implies the existence of a cycle cover of length at most 1.6m. They also showed that the existence of a Fano-flow using at most 4 lines of the Fano plane on bridgeless cubic graphs, which is a consequence of the Fulkerson Conjecture, implies the existence of a cycle cover of length at most 14/9 · m ≈ 1.556m.…”

Section: Introductionmentioning

confidence: 99%

Short Cycle Covers on Cubic Graphs by Choosing a 2-Factor

Candráková¹,

Lukoťka²

2016

SIAM J. Discrete Math.

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We show that every bridgeless cubic graph G with m edges has a cycle cover of length at most 1.6m. Moreover, if G does not contain any intersecting circuits of length 5, then G has a cycle cover of length 212/135 · m ≈ 1.570m and if G contains no 5-circuits, then it has a cycle cover of length at most 14/9 · m ≈ 1.556m. To prove our results, we show that each 2-edge-connected cubic graph G on n vertices has a 2-factor containing at most n/10 + f (G) circuits of length 5, where the value of f (G) only depends on the presence of several subgraphs arising from the Petersen graph. As a corollary we get that each 3-edge-connected cubic graph on n vertices has a 2-factor containing at most n/9 circuits of length 5 and each 4-edge-connected cubic graph on n vertices has a 2-factor containing at most n/10 circuits of length 5. arXiv:1509.07430v1 [math.CO]

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“…Itai and Rodeh also conjectured that every 2‐edge‐connected graph with n vertices and m edges has a circuit cover of length

m + n - 1

, which was proved by Fan in 1998. The following conjecture, independently posed by Jaeger (see [, ]) and Alon and Tarsi in 1985, appears to be one of the most fundamental conjectures concerning circuit covers.…”

Section: Introductionmentioning

confidence: 99%

Circuit Covers of Signed Graphs

Máčajová

Raspaud²,

Rollová

et al. 2015

J. Graph Theory

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We introduce the concept of a signed circuit cover of a signed graph. A signed circuit cover is a natural analog of a circuit cover of a graph and is equivalent to a covering of the corresponding signed graphic matroid with circuits. As in the case of graphs, a signed graph has a signed circuit cover only when it admits a nowhere‐zero integer flow. In the present article, we establish the existence of a universal coefficient q∈R such that every signed graph G that admits a nowhere‐zero integer flow has a signed circuit cover of total length at most q·|E(G)|. We show that if G is bridgeless, then q≤9, and in the general case q≤11.

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Short cycle covers of graphs and nowhere-zero flows

Abstract: A shortest cycle cover of a graph G is a family of cycles which together cover all the edges of G and the sum of their lengths is minimum. In this article we present upper bounds to the length of shortest cycle covers, associated with the existence of two types of nowhere-zero

Cited by 10 publications

References 15 publications

1‐Factor and Cycle Covers of Cubic Graphs

1‐Factor and Cycle Covers of Cubic Graphs

Short Cycle Covers on Cubic Graphs by Choosing a 2-Factor

Circuit Covers of Signed Graphs

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