×

zbMATH — the first resource for mathematics

A solution to a colouring problem of P. Erdős. (English) Zbl 0759.05037
The authors prove a conjecture made by Paul Erdős at the Julius Petersen Graph Theory Conference at Hindsgavl in July 1990: Any 4-regular graph on \(3n\) vertices which has a decomposition into a Hamiltonian circuit and \(n\) pairwise vertex-disjoint triangles has chromatic number 3.
As a consequences, such a graph has independence number \(n\), as conjectured by D. Z. Du and D. F. Hsu in 1987.
It was also noted in M. R. Fellows [SIAM J. Discrete Math. 3, No. 2, 206-215 (1990; Zbl 0735.05057)] that Erdős’ conjecture is equivalent to an old conjecture of Schur, that for any partition of the integers into triples there is another partition of the integers into 3 parts such that each part contains a member of each triple, but no pair of consecutive integers.
Thus did the authors kill three birds with one stone. Hopefully the monetary reward customarily offered by Prof. Erdős will be commensurate with the scope of the result.

MSC:
05C15 Coloring of graphs and hypergraphs
05C45 Eulerian and Hamiltonian graphs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] N. Alon and M. Tarsi, Colourings and orientations of graphs, Combinatorica, to appear. · Zbl 0756.05049
[2] Beineke, L.W.; Wilson, R.J., ()
[3] D.Z. Du, D.F. Hsu and F.K. Hwang, Hamiltonian property of consecutive-d digraphs, Comput. Math. Appl., to appear. · Zbl 0789.05040
[4] D.Z. Du and D.F. Hsu, On a combinatorial problem related to network design and optimization, manuscript.
[5] Fellows, M.R., Transversals of vertex partitions of graphs, SIAM J. discrete math., 3, 206-215, (1990) · Zbl 0735.05057
[6] Fleischner, H., Eularian graphs and related topics, part 1, (), Vol. 2, Ann. Discrete Math. 50
[7] Jaeger, F., On the Penrose number of cubic diagrams, Discrete math., 74, 85-97, (1989) · Zbl 0679.05029
[8] West, D.B., Open problems, SIAM discrete math. newsletter, 1, 3, 9-12, (1991)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.