In graph theory, the unproven Erdős–Gyárfás conjecture, made in 1995 by the prolific 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.
Markström's graph
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.
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. 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 (Heckman & Krakovski 2013)
Weaker results relating the degree of a graph to unavoidable sets of cycle lengths are known: there is a set S of lengths, with |S| = O(n0.99), such that every graph with average degree ten or more contains a cycle with its length in S (Verstraëte 2005), and every graph whose average degree is exponential in the iterated logarithm of n necessarily contains a cycle whose length is a power of two (Sudakov & Verstraëte 2008). The conjecture is also known to be true for planar claw-free graphs (Daniel & Shauger 2001) and for graphs that avoid large induced stars and satisfy additional constraints on their degrees (Shauger 1998).
References
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.
Heckman, Christopher Carl; Krakovski, Roi (2013), "Erdös-Gyárfás conjecture for cubic planar graphs", Electronic Journal of Combinatorics, 20 (2), P7, doi:10.37236/3252.
Markström, Klas (2004), "Extremal graphs for some problems on cycles in graphs" (PDF), Congr. Numerantium, 171: 179–192.
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
Sudakov, Benny; Verstraëte, Jacques (2008), "Cycle lengths in sparse graphs", Combinatorica, 28 (3): 357–372, arXiv:0707.2117, doi:10.1007/s00493-008-2300-6, S2CID 3253855.
Verstraëte, Jacques (2005), "Unavoidable cycle lengths in graphs", Journal of Graph Theory, 49 (2): 151–167, doi:10.1002/jgt.20072, S2CID 247671371.
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
May 05, 2023
erdős, gyárfás, conjecture, unsolved, problem, mathematics, must, every, cubic, graph, contain, simple, cycle, length, power, more, unsolved, problems, mathematics, graph, theory, unproven, made, 1995, prolific, mathematician, paul, erdős, collaborator, andrás. Unsolved problem in mathematics Must every cubic graph contain a simple cycle of length a power of two more unsolved problems in mathematics In graph theory the unproven Erdos Gyarfas conjecture made in 1995 by the prolific mathematician Paul Erdos and his collaborator Andras Gyarfas states that every graph with minimum degree 3 contains a simple cycle whose length is a power of two Erdos offered a prize of 100 for proving the conjecture or 50 for a counterexample it is one of many conjectures of Erdos Markstrom s graphMarkstrom s 24 vertex cubic planar graph with no 4 or 8 cycles found in a computer search for counterexamples to the Erdos Gyarfas conjecture It has however cycles with 16 vertices Vertices24Edges36Radius5Diameter6Girth3Automorphisms3Table of graphs and parametersIf 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 Markstrom that any counterexample must have at least 17 vertices and any cubic counterexample must have at least 30 vertices Markstrom s searches found four graphs on 24 vertices in which the only power of two cycles have 16 vertices One of these four graphs is planar however the Erdos Gyarfas conjecture is now known to be true for the special case of 3 connected cubic planar graphs Heckman amp Krakovski 2013 Weaker results relating the degree of a graph to unavoidable sets of cycle lengths are known there is a set S of lengths with S O n0 99 such that every graph with average degree ten or more contains a cycle with its length in S Verstraete 2005 and every graph whose average degree is exponential in the iterated logarithm of n necessarily contains a cycle whose length is a power of two Sudakov amp Verstraete 2008 The conjecture is also known to be true for planar claw free graphs Daniel amp Shauger 2001 and for graphs that avoid large induced stars and satisfy additional constraints on their degrees Shauger 1998 References EditDaniel Dale Shauger Stephen E 2001 A result on the Erdos Gyarfas conjecture in planar graphs Proc 32nd Southeastern Int Conf Combinatorics Graph Theory and Computing pp 129 139 Heckman Christopher Carl Krakovski Roi 2013 Erdos Gyarfas conjecture for cubic planar graphs Electronic Journal of Combinatorics 20 2 P7 doi 10 37236 3252 Markstrom Klas 2004 Extremal graphs for some problems on cycles in graphs PDF Congr Numerantium 171 179 192 Shauger Stephen E 1998 Results on the Erdos Gyarfas conjecture in K1 m free graphs Proc 29th Southeastern Int Conf Combinatorics Graph Theory and Computing pp 61 65 Sudakov Benny Verstraete Jacques 2008 Cycle lengths in sparse graphs Combinatorica 28 3 357 372 arXiv 0707 2117 doi 10 1007 s00493 008 2300 6 S2CID 3253855 Verstraete Jacques 2005 Unavoidable cycle lengths in graphs Journal of Graph Theory 49 2 151 167 doi 10 1002 jgt 20072 S2CID 247671371 External links EditExoo Geoffrey Graphs Without Cycles of Specified Lengths West Douglas B Erdos Gyarfas Conjecture on 2 power Cycle Lengths Open Problems Graph Theory and Combinatorics Retrieved from https en wikipedia org w index php title Erdos Gyarfas conjecture amp oldid 1136115141, wikipedia, wiki, book, books, library,
