Short Cycle Covers of Graphs with Minimum Degree Three

Kaiser, Tomáš; Kráľ, Daniel; Lidický, Bernard; Nejedlý, Pavel; Šámal, Robert

doi:10.1137/080717468

Cited by 19 publications

(20 citation statements)

References 23 publications

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“…A similar statement holds also for regular nonbipartite graphs provided they are sufficiently edge-connected. This statement has found applications in many areas of graph theory (see, e.g., [1,3]).…”

Section: Introductionmentioning

confidence: 90%

Perfect Matchings of Regular Bipartite Graphs

Lukoťka

Rollová

2016

Journal of Graph Theory

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Let G be a regular bipartite graph and X⊆E(G). We show that there exist perfect matchings of G containing both, an odd and an even number of edges from X if and only if the signed graph (G,X), that is a graph G with exactly the edges from X being negative, is not equivalent to (G,∅). In fact, we prove that for a given signed regular bipartite graph with minimum signature, it is possible to find perfect matchings that contain exactly no negative edges or an arbitrary one preselected negative edge. Moreover, if the underlying graph is cubic, there exists a perfect matching with exactly two preselected negative edges. As an application of our results we show that each signed regular bipartite graph that contains an unbalanced circuit has a 2‐cycle‐cover such that each cycle contains an odd number of negative edges.

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Section: Introductionmentioning

confidence: 90%

Perfect Matchings of Regular Bipartite Graphs

Lukoťka

Rollová

2016

Journal of Graph Theory

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“…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. Recently, Kaiser et al [12] proved that every bridgeless cubic graph G has a cycle cover of length at most 34 21 |E(G)| and that any bridgeless graph G of minimum degree three has a cycle cover of length at most 44 27 |E(G)|. SCCP is closely related to various kinds of flow problems.…”

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

confidence: 98%

Short cycle covers of graphs and nowhere-zero flows

et al. 2011

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“… show that

scc (G) \leq 8 | E (G) | / 5

. With additional information on cycle or 2‐factor structures, some upper bounds on shortest circuit cover of cubic graphs are obtained in . In general, Alon and Tarsi made the following conjecture – the Shortest Circuit Cover Conjecture.…”

Section: Introductionmentioning

confidence: 99%

Circuit covers of cubic signed graphs

2018

Journal of Graph Theory

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A signed graph, denoted by (G,σ), is a graph G associated with a mapping σ:E(G)→{−1,+1}. A cycle of (G,σ) is a connected 2‐regular subgraph. A cycle C is positive if it has an even number of negative edges, and negative otherwise. A signed‐circuit of a signed graph (G,σ) is a positive cycle or a barbell consisting of two edge‐disjoint negative cycles joined by a path. The definition of a signed‐circuit of signed graph comes from the signed‐graphic matroid. A signed‐circuit cover of (G,σ) is a family of signed‐circuits covering all edges of (G,σ). A signed‐circuit cover with the smallest total length is called a shortest signed‐circuit cover of (G,σ) and its length is denoted by scc(G,σ). Bouchet proved that a signed graph has a signed‐circuit cover if and only if it is flow‐admissible (i.e., has a nowhere‐zero integer flow). In this article, we show that a 3‐connected flow‐admissible signed graph does not necessarily have a signed‐circuit double cover. For shortest signed‐circuit cover of 2‐edge‐connected cubic signed graphs (G,σ), we show that scc(G,σ)<26|E(G)|/9 if it is flow‐admissible.

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Short Cycle Covers of Graphs with Minimum Degree Three

Cited by 19 publications

References 23 publications

Perfect Matchings of Regular Bipartite Graphs

Perfect Matchings of Regular Bipartite Graphs

Short cycle covers of graphs and nowhere-zero flows

Circuit covers of cubic signed graphs

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