A proof of the Erdös-Faber-Lovász conjecture: Algorithmic aspects

Kang, Donghoon; Kelly, Tom; Kühn, Daniela; Methuku, Abhishek; Osthus, Deryk

doi:10.1109/focs52979.2021.00107

Cited by 10 publications

(15 citation statements)

References 51 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, as in the previous section, if one can show that for every possible clique-decomposition D of K n , (K n , D) admits a n-colouring, then the EFL Conjecture would be true. Although a proof of the conjecture for all sufficiently large values of n was recently announced [7], we still believe that such a problem deserves to be studied further and solved for the other instances as well, as this could give insights into related areas such as clique-decompositions and edge-colourings of graphs, which have been already studied such as in [3,9]. In this sense, we suggest the following problem which we think could be a possible way forward.…”

Section: Resultsmentioning

confidence: 89%

See 1 more Smart Citation

The Erdős--Faber--Lovász Conjecture revisited

Gauci,

Zerafa

2021

Preprint

View full text Add to dashboard Cite

The Erdős-Faber-Lovász Conjecture, posed in 1972, states that if a graph G is the union of n cliques of order n (referred to as defining n-cliques) such that two cliques can share at most one vertex, then the vertices of G can be properly coloured using n colours. Although still open after almost 50 years, it can be easily shown that the conjecture is true when every shared vertex belongs to exactly two defining n-cliques. We here provide a quick and easy algorithm to colour the vertices of G in this case, and discuss connections with clique-decompositions and edge-colourings of graphs.

show abstract

Section: Resultsmentioning

confidence: 89%

“…In particular, we remark that the fractional version of the EFL Conjecture was solved by Kahn and Seymour [6] in 1992. Moreover, in January 2021, it was announced [7] that the conjecture is true for sufficiently large values of n, which to our knowledge, is the best result so far in trying to attack the EFL Conjecture.…”

Section: Introductionmentioning

confidence: 80%

The Erdős--Faber--Lovász Conjecture revisited

Gauci,

Zerafa

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The absorbing method was initially explicitly introduced by Rödl, Ruciński, and Szemerédi [61] even though the implicit idea has its roots in the works of Krivelevich [44] and Erdős, Gyárfás, and Pyber [16]. Recently it has seen a surge of interest and has been used in a variety of settings: combinatorial designs [27,35], decompositions [26,48], Steiner systems [50,19], Ramsey theory [9,37], colouring (hyper)graphs [54,33], embeddings [53,24], and many, many more.…”

Section: The Absorbing Methods In Sparse Hypergraphsmentioning

confidence: 99%

Transference for loose Hamilton cycles in random $3$-uniform hypergraphs

Kalina¹,

Trujić²

2022

Preprint

View full text Add to dashboard Cite

A loose Hamilton cycle in a hypergraph is a cyclic sequence of edges covering all vertices in which only every two consecutive edges intersect and do so in exactly one vertex. With Dirac's theorem in mind, it is natural to ask what minimum d-degree condition guarantees the existence of a loose Hamilton cycle in a k-uniform hypergraph. For k " 3 and each d P t1, 2u, the necessary and sufficient such condition is known precisely. We show that these results adhere to a 'transference principle' to their sparse random analogues. The proof combines several ideas from the graph setting and relies on the absorbing method. In particular, we employ a novel approach of Kwan and Ferber for finding rooted absorbers in subgraphs of sparse hypergraphs via a contraction procedure. In the case of d " 2, our findings are asymptotically optimal.

show abstract

“…A special case of a well-known conjecture of Erdös, Faber and Lovász [11] can be formulated as the statement that the chromatic index of any S (2, k, v) is no more than v. This conjecture has been showed to hold for any cyclic S(2, k, v) by Colbourn and Colbourn [9], and for any S(2, k, v) with sufficiently large v by Kang, Kelly, Kühn, Methuku and Osthus [16].…”

Section: Introductionmentioning

confidence: 98%

The chromatic index of finite projective spaces

Xu¹,

Feng²

2022

Preprint

View full text Add to dashboard Cite

A coloring of PG(n, q), the n-dimensional projective geometry over GF(q), is an assignment of colors to all lines of PG(n, q) so that any two lines with the same color do not intersect. The chromatic index of PG(n, q), denoted by χ ′ (PG(n, q)), is the least number of colors for which a coloring of PG(n, q) exists. This paper translates the problem of determining the chromatic index of PG(n, q) to the problem of examining the existences of PG(3, q) and PG(4, q) with certain properties. As applications, it is shown that for any odd integer n and q ∈ {3, 4, 8}, χ ′ (PG(n, q)) = n 1 q , which implies the existence of a parallelism of PG(n, q) for any odd integer n and q ∈ {3, 4, 8}.

show abstract

A proof of the Erdös-Faber-Lovász conjecture: Algorithmic aspects

Cited by 10 publications

References 51 publications

The Erdős--Faber--Lovász Conjecture revisited

The Erdős--Faber--Lovász Conjecture revisited

Transference for loose Hamilton cycles in random $3$-uniform hypergraphs

The chromatic index of finite projective spaces

Contact Info

Product

Resources

About