Erdős–Gyárfás conjecture

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.

Vertices24

Edges36

Radius5

Diameter6

Quick facts Markström's graph, Vertices ...

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.
Vertices	24
Edges	36
Radius	5
Diameter	6
Girth	3
Automorphisms	3
Table of graphs and parameters

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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).

The conjecture remains open for bipartite cubic graphs. This case has been approached through several related computational and structural directions. Royle's searches established the 17-vertex lower bound for unrestricted counterexamples, while Markström's exhaustive searches among cubic graphs raised the corresponding bound to 30 vertices (Markström 2004). In work presented at the 14th Workshop on Graph Theory in Szklarska Poręba, Poland, in 2011, Pouria Salehi Nowbandegani and Hossein Esfandiari separately searched the bipartite case and proved that a bipartite counterexample must also have at least 30 vertices (Salehi Nowbandegani & Esfandiari 2011). Consequently, a cubic bipartite counterexample, if one exists, must have at least 30 vertices. 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 (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(n^0.99), 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).