On Cubic Bridgeless Graphs Whose Edge-Set Cannot be Covered by Four Perfect Matchings

Mazzuoccolo, Giuseppe

doi:10.1002/jgt.21778

Cited by 21 publications

(7 citation statements)

References 14 publications

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“…Thus the Gyárfás–Sumner conjecture asserts that, for every forest

H

, the class of all

H

‐free graphs is

\chi

‐bounded. Esperet [5] conjectured that every

\chi

‐bounded class is polynomially

\chi

‐ bounded , that is,

f

can be chosen to be a polynomial. Neither conjecture has been settled in general.…”

Section: Introductionmentioning

confidence: 99%

Polynomial bounds for chromatic number II: Excluding a star‐forest

Scott

Seymour

Spirkl

2022

Journal of Graph Theory

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The Gyárfás–Sumner conjecture says that for every forest H $H$, there is a function f H ${f}_{H}$ such that if G $G$ is H $H$‐free then χ ( G ) ≤ f H ( ω ( G ) ) $\chi (G)\le {f}_{H}(\omega (G))$ (where χ , ω $\chi ,\omega $ are the chromatic number and the clique number of G $G$). Louis Esperet conjectured that, whenever such a statement holds, f H ${f}_{H}$ can be chosen to be a polynomial. The Gyárfás–Sumner conjecture is only known to be true for a modest set of forests H $H$, and Esperet's conjecture is known to be true for almost no forests. For instance, it is not known when H $H$ is a five‐vertex path. Here we prove Esperet's conjecture when each component of H $H$ is a star.

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“…Thus the Gyárfás–Sumner conjecture asserts that, for every forest

H

, the class of all

H

‐free graphs is

\chi

‐bounded. Esperet [5] conjectured that every

\chi

‐bounded class is polynomially

\chi

‐ bounded , that is,

f

can be chosen to be a polynomial. Neither conjecture has been settled in general.…”

Section: Introductionmentioning

confidence: 99%

Polynomial bounds for chromatic number II: Excluding a star‐forest

Scott

Seymour

Spirkl

2022

Journal of Graph Theory

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“…In this case, every three perfect matchings from a cover of this type form an FR-triple since every edge has frequency one or two with respect to this cover. Therefore, a possible counterexample to the Fan-Raspaud Conjecture should be searched for in the class of bridgeless cubic graphs whose edge-set cannot be covered by four perfect matchings, see for instance [6]. In 2009, Máčajová and Škoviera [22] shed some light on the Fan-Raspaud Conjecture by proving it for bridgeless cubic graphs having oddness two.…”

Section: Further Results On the S 4 -Conjecturementioning

confidence: 99%

An equivalent formulation of the Fan-Raspaud Conjecture and related problems

Mazzuoccolo

Zerafa²

2020

Ars Math. Contemp.

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In 1994, it was conjectured by Fan and Raspaud that every simple bridgeless cubic graph has three perfect matchings whose intersection is empty. In this paper we answer a question recently proposed by Mkrtchyan and Vardanyan, by giving an equivalent formulation of the Fan-Raspaud Conjecture. We also study a possibly weaker conjecture originally proposed by the first author, which states that in every simple bridgeless cubic graph there exist two perfect matchings such that the complement of their union is a bipartite graph. Here, we show that this conjecture can be equivalently stated using a variant of Petersen-colourings, we prove it for graphs having oddness at most four and we give a natural extension to bridgeless cubic multigraphs and to certain cubic graphs having bridges.

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“…Finally, when H = P 12 , Conjecture 3.2 implies that H-problem is equivalent to testing whether the input graph G can be covered with four perfect matchings. The latter problem is proved to be NP-complete in [3]. Thus, depending on the choice of H, the H-problem may or may not be NP-complete.…”

Section: Future Workmentioning

confidence: 99%

S_12 and P_12-colorings of cubic graphs

Hakobyan¹,

Mkrtchyan²

2019

Ars Math. Contemp.

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If G and H are two cubic graphs, then an H-coloring of G is a proper edge-coloring f with the edges of H, such that for each vertex x of G, there is a vertex y of H with f (∂ G (x)) = ∂ H (y). If G admits an H-coloring, then we will write H ≺ G. The Petersen coloring conjecture of Jaeger (P 10-conjecture) states that for any bridgeless cubic graph G, one has: P 10 ≺ G. The S 10-conjecture states that for any cubic graph G, S 10 ≺ G. In this paper, we introduce two new conjectures that are related to these conjectures. The first of them states that any cubic graph with a perfect matching admits an S 12-coloring. The second one states that any cubic graph G whose edge-set can be covered with four perfect matchings, admits a P 12-coloring. We call these new conjectures S 12-conjecture and P 12conjecture, respectively. Our first results justify the choice of graphs in S 12-conjecture and P 12-conjecture. Next, we characterize the edges of P 12 that may be fictive in a P 12coloring of a cubic graph G. Finally, we relate the new conjectures to the already known conjectures by proving that S 12-conjecture implies S 10-conjecture, and P 12-conjecture and (5, 2)-Cycle cover conjecture together imply P 10-conjecture. Our main tool for proving the latter statement is a new reformulation of (5, 2)-Cycle cover conjecture, which states that the edge-set of any claw-free bridgeless cubic graph can be covered with four perfect matchings.

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On Cubic Bridgeless Graphs Whose Edge-Set Cannot be Covered by Four Perfect Matchings

Cited by 21 publications

References 14 publications

Polynomial bounds for chromatic number II: Excluding a star‐forest

Polynomial bounds for chromatic number II: Excluding a star‐forest

An equivalent formulation of the Fan-Raspaud Conjecture and related problems

S_12 and P_12-colorings of cubic graphs

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