User:MWinter4/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 has the $$P_k$$-property (and is said to be a $$P_k$$-graph) if each pair of its vertices is connected by exactly one path of length $$k$$. Kotzig's conjecture asserts that for $$k\ge 3$$ there are no $$P_k$$-graphs with two or more vertices. The conjecture was first formulated by Anton Kotzig in 1974, who proved it for $$k\le 8$$. It was later verified for $$k\le 20$$ by Alexandr Kostochka, yet remains open in the general case (as of July 2024).

There do exist $$P_k$$-graphs for $$k\le 2$$. $$P_0$$-graphs are exactly the graphs without edges; and $$P_1$$-graphs exactly the complete graphs. The friendship theorem states that $$P_2$$-graphs are exactly the (triangular) windmill graphs (finitely many triangles joined at a common vertex; also known as friendship graphs).

History
Kotzig's conjecture has an unfortunatey history in that the literature contains claims of its resolution, while in actuality it is open.

Kotzig himself verified the conjecture for $$k\le 8$$ (1979). In 1990 Xing and Hu published a paper, claiming to verify Kotzig's conjecture for $$k\ge 12$$ (with the gap later filled independently by Yang, Lin, Wang, Li for $$9\le k\le 11$$; and Kostochka for $$k\le 20$$). This seemed to resolve the conjecture at the time. However, Xing and Hu's proof relied on a misunderstanding in a statement proven by Kotzig. Kotzig proved that a $$P_k$$-graph must contain a $$2\ell$$-cycle for some $$\ell\in\{3,...,k-4\}$$, which Xing and Hu's state correctly, but use in the form that all these cycles exist. More specifically, they show that for $$k\ge 12$$ a $$P_k$$-graph must contain a $$(2k-8)$$-cycle. Since this is in contradiction to their reading of Kotzig's result, they conclude $$P_k$$-graphs cannot exist. This mistake was first pointed out by Roland Häggvist in 2000.

Kotzig's conjecture is mentioned in Proofs from THE BOOK (a collection of proofs by M. Aigner and G. M. Ziegler) in the chapter on the friendship theorem. It is stated that a general proof for the conjecture seems "out of reach".

The conjecture is known to be true for $$k\le 20$$ due to work of Alexandr Kostochka. He furthermore claims the conjecture to be true for $$k\le 33$$, but a proof has neven been published.

Properties of $$P_k$$-graphs

 * A $$P_k$$-graph on $$n$$ vertices contains precisely $$\textstyle {n\choose 2}$$ paths of length $$k$$.


 * Since the two end-vertices of an edge in a $$P_k$$-graph are connected by a unique $$k$$-path, each edge is contained in a unique $$(k+1)$$-cycle. Consequently, the graph is an edge disjoint union of $$(k+1)$$-cycles, and there are no other $$(k+1)$$-cycles besides these. In particular, $$P_k$$-graphs are Eulerian.


 * $$P_k$$-graphs are not bipartite: if $$k$$ is odd and $$v,w$$ are vertices in the same bipartition class, no $$k$$-path can connect them. Likewise, if $$k$$ is even and $$v,w$$ are vertices in different bipartition classes, no $$k$$-path can connect them.


 * Even cycles are critical substructures in $$P_k$$-graphs as they can form so-called lollipops (or monocles): the union of an even $$2\ell$$-cycle and a path that intersects the cycles precisely in one end vertex. The path must be shorter than $$k-\ell$$ as it would otherwise give rise to a forbidden double-$$k$$-path. Therefore, the existence and structure of even cycles as been studied extensively. It is known that there cannot be an even cycle of length $$4$$, $$2k$$, $$2k-2$$, $$2k-4$$, $$2k-6$$ (Kotzig) and $$2k-8$$ (Xing, Hu).


 * Kostochka proved that a $$P_k$$-graph cannot contain a cycle (even or odd) of length at least $$\tfrac43 k-2$$. Kostochka furthermore showed that there must exist a cycle of length at least $$k+5$$. Combining these constraints can be rearranged to yield $$k\ge 21$$.


 * Any two $$(k+1)$$-cycles in a $$P_k$$-graph must have at least three and at most $$k-2$$ vertices in common. In particular, $$G$$ is 2-connected. (Kotzig / Bondy / Kostochka)


 * If $$c_{k+1}$$ denotes the number of $$(k+1)$$-cycles in a given $$P_k$$-graph, then $$c_{k+1}\ge 3$$, and $$c_{k+1}\ge 4$$ if $$k$$ is even (Kostochka). If $$k$$ is odd, then $$c_{k+1}\le \tfrac12(k-1)$$ (Bondy). Consequently, a $$P_k$$-graph for $$k$$ odd can have at most $$\tfrac12(k-1)(k+1)+1$$ vertices.