Covering Multigraphs by Simple Circuits

Alon, Noga; Tarsi, Michael

doi:10.1137/0606035

Cited by 70 publications

(73 citation statements)

References 7 publications

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“…Since M is a 1-factor of G c , it follows that each circuit contains at least 1 2 girth(G c ) edges of U. Hence, there are at most 2k girth(Gc) pairwise disjoint circuits in G c .…”

Section: Conjecture 15 ([1])mentioning

confidence: 99%

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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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“…Since M is a 1-factor of G c , it follows that each circuit contains at least 1 2 girth(G c ) edges of U. Hence, there are at most 2k girth(Gc) pairwise disjoint circuits in G c .…”

Section: Conjecture 15 ([1])mentioning

confidence: 99%

“…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

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“…Then G admits a nowhere-zero 4-flow which implies scc(G) ≤ 3 . But this is true for any bridgeless graph by Alon and Tarsi [1] and Bermond et al [2]. We conclude that the conjecture is also true for any graph with c (G) = 6.…”

Section: Shortest Cycle Covers and Circular Flowsmentioning

confidence: 63%

“…The best approximation of this conjecture for general bridgeless graphs was obtained by Bermond et al [2] and independently by Alon and Tarsi [1], who proved that every bridgeless graph G has a cycle cover of length at most 5 3 |E(G)|. It follows from the Splitting Lemma of Fleischner [6] that Conjecture 1.1 is equivalent to its restriction to the family of bridgeless subcubic graphs.…”

Section: Conjecture 11 Every Bridgeless Graph G Has a Cycle Cover Omentioning

confidence: 94%

Short cycle covers of graphs and nowhere-zero flows

et al. 2011

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“…The circuit cover problem is related to problems involving graph embeddings [Arc, Hag, Lit, Tut], flow theory [Cel,Fan2,Jael,You], short circuit covers [Alo,Ber,Fanl,Gua,Jac,Jam2,Jam3,Tari,Zhal], the Chinese Postman Problem [Edm, Gua, Ita, Jac], perfect matchings [Ful,God2,p. 22] and decompositions of eulerian graphs [Fiel,Fle2,Sey3].…”

Section: Introductionmentioning

confidence: 99%

Graphs with the Circuit Cover Property

Alspach¹,

Goddyn²,

Zhang³

1994

Transactions of the American Mathematical Society

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Abstract. A circuit cover of an edge-weighted graph (G, p) is a multiset of circuits in G such that every edge e is contained in exactly p(e) circuits in the multiset. A nonnegative integer valued weight vector p is admissible if the total weight of any edge-cut is even, and no edge has more than half the total weight of any edge-cut containing it. A graph G has the circuit cover property if (G, p) has a circuit cover for every admissible weight vector p . We prove that a graph has the circuit cover property if and only if it contains no subgraph homeomorphic to Petersen's graph. In particular, every 2-edge-connected graph with no subgraph homeomorphic to Petersen's graph has a cycle double cover.

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Covering Multigraphs by Simple Circuits

Cited by 70 publications

References 7 publications

1‐Factor and Cycle Covers of Cubic Graphs

1‐Factor and Cycle Covers of Cubic Graphs

Short cycle covers of graphs and nowhere-zero flows

Graphs with the Circuit Cover Property

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