Erdős–Gyárfás conjecture
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Unsolved problem in mathematics: Must every cubic graph contain a simple cycle of length a power of two? (more unsolved problems in mathematics)
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| Markström's graph | |
|---|---|
| File:220px Markström's 24-vertex cubic planar graph with no 4- or 8-cycles, found in a computer search for counterexamples to the Erdős–Gyárfás conjecture. It has, however, cycles with 16 vertices. | |
| Vertices | 24 |
| Edges | 36 |
| Radius | 5 |
| Diameter | 6 |
| Girth | 3 |
| Automorphisms | 3 |
| Table of graphs and parameters | |
In graph theory, the unproven Erdős–Gyárfás conjecture, made in 1995 by mathematician Paul Erdős and his collaborator András Gyárfás, states that every graph with minimum degree 3 contains a simple cycle whose length is a power of two. Erdős offered a prize of $100 for proving the conjecture, or $50 for a counterexample; it is one of many conjectures of Erdős.
If the conjecture is false, a counterexample would take the form of a graph with minimum degree three having no power-of-two cycles. It is known through computer searches of Gordon Royle and Klas Markström that any counterexample must have at least 17 vertices, and any cubic counterexample must have at least 30 vertices. Markström's searches found four graphs on 24 vertices in which the only power-of-two cycles have 16 vertices.[1] One of these four graphs is planar; however, the Erdős–Gyárfás conjecture is now known to be true for the special case of 3-connected cubic planar graphs.[2]
The conjecture remains open for bipartite cubic graphs. This case has been approached through several related computational and structural directions. In work presented at the 14th Workshop on Graph Theory in Szklarska Poręba, Poland, in 2011, Pouria Salehi Nowbandegani and Hossein Esfandiari proved by computer search that a cubic bipartite counterexample must have at least 30 vertices.[3] In 2026, Julius Tranquilli improved this lower bound to 60 vertices, using a certified exhaustive computation to show that every simple cubic bipartite graph on at most 58 vertices contains a cycle of length 4, 8, or 16.[4] Consequently, any cubic bipartite counterexample to the Erdős–Gyárfás conjecture must have at least 60 vertices. The computation was independently checked using two exact procedures with different 16-cycle detection methods and a static witness certificate.[4] The 2013 result of Heckman and Krakovski settles a different but overlapping cubic restriction—3-connected cubic planar graphs—but does not settle the general cubic bipartite case.[2]
Weaker results relating the degree of a graph to unavoidable sets of cycle lengths are known: there is a set of lengths, with , such that every graph with average degree ten or more contains a cycle with its length in ,[5] and every graph whose average degree is exponential in the iterated logarithm of necessarily contains a cycle whose length is a power of two.[6] The conjecture is also known to be true for planar claw-free graphs[7] and for graphs that avoid large induced stars and satisfy additional constraints on their degrees.[8]
References
- ↑ Markström, Klas (2004), "Extremal graphs for some problems on cycles in graphs", Congr. Numerantium 171: 179–192, http://abel.math.umu.se/~klasm/Uppsatser/cycex.pdf
- ↑ 2.0 2.1 Heckman, Christopher Carl; Krakovski, Roi (2013), "Erdös-Gyárfás conjecture for cubic planar graphs", Electronic Journal of Combinatorics 20 (2): P7, doi:10.37236/3252, http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i2p7
- ↑ Salehi Nowbandegani, Pouria; Esfandiari, Hossein (2011), "An experimental result on the Erdős–Gyárfás conjecture in bipartite graphs", Proceedings of the 14th Workshop on Graph Theory: Colourings, Independence and Domination, Szklarska Poręba, Poland, pp. 18–23
- ↑ 4.0 4.1 Tranquilli, Julius (2026), A 60-Vertex Lower Bound for Cubic Bipartite Counterexamples to the Erdős–Gyárfás Conjecture
- ↑ Verstraëte, Jacques (2005), "Unavoidable cycle lengths in graphs", Journal of Graph Theory 49 (2): 151–167, doi:10.1002/jgt.20072
- ↑ Sudakov, Benny; Verstraëte, Jacques (2008), "Cycle lengths in sparse graphs", Combinatorica 28 (3): 357–372, doi:10.1007/s00493-008-2300-6
- ↑ Daniel, Dale; Shauger, Stephen E. (2001), "A result on the Erdős–Gyárfás conjecture in planar graphs", Proc. 32nd Southeastern Int. Conf. Combinatorics, Graph Theory, and Computing, pp. 129–139
- ↑ Shauger, Stephen E. (1998), "Results on the Erdős–Gyárfás conjecture in K1,m-free graphs", Proc. 29th Southeastern Int. Conf. Combinatorics, Graph Theory, and Computing, pp. 61–65
External links
- Exoo, Geoffrey, Graphs Without Cycles of Specified Lengths
- West, Douglas B., Erdős Gyárfás Conjecture on 2-power Cycle Lengths, Open Problems - Graph Theory and Combinatorics
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