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Even circuit theorem

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In extremal graph theory, the even circuit theorem is a result of Paul Erdős according to which an n-vertex graph that does not have a simple cycle of length 2k can only have O(n1 + 1/k) edges. For instance, 4-cycle-free graphs have O(n3/2) edges, 6-cycle-free graphs have O(n4/3) edges, etc.

History

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The result was stated without proof by Erdős in 1964.[1] Bondy & Simonovits (1974) published the first proof, and strengthened the theorem to show that, for n-vertex graphs with Ω(n1 + 1/k) edges, all even cycle lengths between 2k and 2kn1/k occur.[2]

Lower bounds

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Unsolved problem in mathematics:
Do there exist -cycle-free graphs (for other than , , or ) that have edges?

The bound of Erdős's theorem is tight up to constant factors for some small values of k: for k = 2, 3, or 5, there exist graphs with Ω(n1 + 1/k) edges that have no 2k-cycle.[2][3][4]

It is unknown for k other than 2, 3, or 5 whether there exist graphs that have no 2k-cycle but have Ω(n1 + 1/k) edges, matching Erdős's upper bound.[5] Only a weaker bound is known, according to which the number of edges can be Ω(n1 + 2/(3k − 3)) for odd values of k, or Ω(n1 + 2/(3k − 4)) for even values of k.[4]

Constant factors

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Because a 4-cycle is a complete bipartite graph, the maximum number of edges in a 4-cycle-free graph can be seen as a special case of the Zarankiewicz problem on forbidden complete bipartite graphs, and the even circuit theorem for this case can be seen as a special case of the Kővári–Sós–Turán theorem. More precisely, in this case it is known that the maximum number of edges in a 4-cycle-free graph is

Erdős & Simonovits (1982) conjectured that, more generally, the maximum number of edges in a 2k-cycle-free graph is

[6]

However, later researchers found that there exist 6-cycle-free graphs and 10-cycle-free graphs with a number of edges that is larger by a constant factor than this conjectured bound, disproving the conjecture. More precisely, the maximum number of edges in a 6-cycle-free graph lies between the bounds

where ex(n,G) denotes the maximum number of edges in an n-vertex graph that has no subgraph isomorphic to G.[3] The maximum number of edges in a 10-cycle-free graph can be at least[4]

The best proven upper bound on the number of edges, for 2k-cycle-free graphs for arbitrary values of k, is

[5]

References

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  1. ^ Erdős, P. (1964), "Extremal problems in graph theory" (PDF), Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963), Publ. House Czechoslovak Acad. Sci., Prague, pp. 29–36, MR 0180500.
  2. ^ a b Bondy, J. A.; Simonovits, M. (1974), "Cycles of even length in graphs" (PDF), Journal of Combinatorial Theory, Series B, 16: 97–105, doi:10.1016/0095-8956(74)90052-5, MR 0340095.
  3. ^ a b Füredi, Zoltan; Naor, Assaf; Verstraëte, Jacques (2006), "On the Turán number for the hexagon", Advances in Mathematics, 203 (2): 476–496, doi:10.1016/j.aim.2005.04.011, MR 2227729.
  4. ^ a b c Lazebnik, F.; Ustimenko, V. A.; Woldar, A. J. (1994), "Properties of certain families of 2k-cycle-free graphs", Journal of Combinatorial Theory, Series B, 60 (2): 293–298, doi:10.1006/jctb.1994.1020, MR 1271276.
  5. ^ a b Pikhurko, Oleg (2012), "A note on the Turán function of even cycles" (PDF), Proceedings of the American Mathematical Society, 140 (11): 3687–3692, doi:10.1090/S0002-9939-2012-11274-2, MR 2944709.
  6. ^ Erdős, P.; Simonovits, M. (1982), "Compactness results in extremal graph theory" (PDF), Combinatorica, 2 (3): 275–288, doi:10.1007/BF02579234, MR 0698653, archived from the original (PDF) on 2016-03-04, retrieved 2015-11-06.