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- 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. 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 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, 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. The conjecture is also known to be true for planar claw-free graphs and for graphs that avoid large induced stars and satisfy additional constraints on their degrees. (en)
- En théorie des graphes, la conjecture d' Erdős-Gyárfás, formulée en 1995 par les mathématiciens Paul Erdős et András Gyárfás, est la suivante : Conjecture d'Erdős-Gyárfás — Tout graphe de degré au moins 3 contient un cycle simple dont la longueur est une puissance de deux Erdős a offert un prix de 100 $ pour la preuve de la conjecture, ou 50 $ pour un contre-exemple ; c'est l'une des nombreuses conjectures d'Erdős . (fr)
- In teoria dei grafi, l'indimostrata congettura di Erdős–Gyárfás, proposta nel 1995 dal prolifico matematico Paul Erdős e il suo collaboratore András Gyárfás, afferma che ogni grafo con grado minimo 3 contiene un ciclo semplice la cui lunghezza è una potenza di 2. Erdős mise in palio $100 per la dimostrazione della congettura, o $50 per un controesempio. Grazie alle ricerche al computer di e , è noto che un eventuale controesempio deve avere almeno 17 vertici, e ogni controesempio cubico deve avere almeno 30 vertici. Le ricerche di Markström hanno consentito di trovare quattro grafi con 24 vertici in cui gli unici cicli di lunghezza pari ad una potenza di 2 hanno 16 vertici; uno di questi quattro grafi è planare. (it)
- Гипотеза Эрдёша — Дьярфаша — нерешённая проблема в теории графов (ru)
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- 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. (en)
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- En théorie des graphes, la conjecture d' Erdős-Gyárfás, formulée en 1995 par les mathématiciens Paul Erdős et András Gyárfás, est la suivante : Conjecture d'Erdős-Gyárfás — Tout graphe de degré au moins 3 contient un cycle simple dont la longueur est une puissance de deux Erdős a offert un prix de 100 $ pour la preuve de la conjecture, ou 50 $ pour un contre-exemple ; c'est l'une des nombreuses conjectures d'Erdős . (fr)
- Гипотеза Эрдёша — Дьярфаша — нерешённая проблема в теории графов (ru)
- 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. (en)
- In teoria dei grafi, l'indimostrata congettura di Erdős–Gyárfás, proposta nel 1995 dal prolifico matematico Paul Erdős e il suo collaboratore András Gyárfás, afferma che ogni grafo con grado minimo 3 contiene un ciclo semplice la cui lunghezza è una potenza di 2. Erdős mise in palio $100 per la dimostrazione della congettura, o $50 per un controesempio. (it)
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- Erdős–Gyárfás conjecture (en)
- Conjecture d'Erdős-Gyárfás (fr)
- Congettura di Erdős-Gyárfás (it)
- Гипотеза Эрдёша — Дьярфаша (ru)
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