Kotzig's conjecture

Kotzig's conjecture is an unproven assertion in graph theory which states that finite graphs with certain properties do not exist. A graph is a P k {\displaystyle P_{k}} -graph if each pair of distinct vertices is connected by exactly one path of length k {\displaystyle k} . Kotzig's conjecture asserts that for k ≥ 3 {\displaystyle k\geq 3} there are no finite P k {\displaystyle P_{k}} -graphs with two or more vertices. The conjecture was first formulated by Anton Kotzig in 1974. It has been verified for k ≤ 20 {\displaystyle k\leq 20} , but remains open in the general case (as of July 2026). The conjecture is stated for k ≥ 3 {\displaystyle k\geq 3} because P k {\displaystyle P_{k}} -graphs do exist for smaller values of k {\displaystyle k} . P 1 {\displaystyle P_{1}} -graphs are precisely the complete graphs. The friendship theorem states that P 2 {\displaystyle P_{2}} -graphs are precisely the (triangular) windmill graphs (that is, finitely many triangles joined at a common vertex; also known as friendship graphs).

History Kotzig's conjecture was first listed as an open problem by Bondy & Murty in 1976, attributed to Kotzig and dated to 1974. Kotzig's first own writing on the conjecture appeared in 1979. He later verified the conjecture for k ≤ 8 {\displaystyle k\leq 8} and claimed solution, though unpublished, for k ≤ 9 {\displaystyle k\leq 9} . The conjecture is now known to hold for k ≤ 20 {\displaystyle k\leq 20} due to work of Alexandr Kostochka. Kostochka stated that his techniques extend to k ≤ 33 {\displaystyle k\leq 33} , but a proof of this has not been published. A survey on P k {\displaystyle P_{k}} -graphs was written by John A. Bondy, including proofs for many statements previously made by Kotzig without written proof. In 1990 Xing & Hu claimed a proof of Kotzig's conjecture for k ≥ 12 {\displaystyle k\geq 12} . This seemed to resolve the conjecture at the time, and still today leads many to believe that the problem is settled. However, Xing and Hu's proof relied on a misunderstanding of a statement proven by Kotzig. Kotzig showed that a P k {\displaystyle P_{k}} -graph must contain a 2 ℓ {\displaystyle 2\ell } -cycle for some ℓ ∈ { 3 , . . . , k − 4 } {\displaystyle \ell \in \{3,...,k-4\}} , which Xing and Hu used in the form that cycles of all these lengths must exist. In their paper Xing and Hu show that for k ≥ 12 {\displaystyle k\geq 12} a P k {\displaystyle P_{k}} -graph must not contain a ( 2 k − 8 ) {\displaystyle (2k-8)} -cycle. Since this is in contradiction to their reading of Kotzig's result, they conclude (incorrectly) that P k {\displaystyle P_{k}} -graphs with k ≥ 12 {\displaystyle k\geq 12} cannot exist. This mistake was first pointed out by Roland Häggkvist in 2000. Kotzig's conjecture is mentioned in Proofs from THE BOOK in the chapter on the friendship theorem. It is stated that a general proof for the conjecture seems "out of reach".

Properties of P k {\displaystyle P_{k}} -graphs A P k {\displaystyle P_{k}} -graph on n {\displaystyle n} vertices contains precisely ( n 2 ) {\displaystyle \textstyle {n \choose 2}} paths of length k {\displaystyle k} . Since the two end-vertices of an edge in a P k {\displaystyle P_{k}} -graph are connected by a unique k {\displaystyle k} -path, each edge is contained in a unique ( k + 1 ) {\displaystyle (k+1)} -cycle. Consequently, the graph has a unique decomposition into edge disjoint ( k + 1 ) {\displaystyle (k+1)} -cycles, and there are no other ( k + 1 ) {\displaystyle (k+1)} -cycles besides these. In particular, P k {\displaystyle P_{k}} -graphs are Eulerian.

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