---
tag: 10FX3BAMEE
kind: conjecture
name: Kotzig's conjecture
aliases: [conjecture-of-kotzig, regular-path-connectedness, regular-path-connectivity, regularly-path-connected-graph]
sources:
  - wikipedia: en:77390020@r1366924136
    locator: Kotzig's conjecture
    note: adopted
    mention: wikipedia://en/77390020
    sections: [Statement]
same_as:
  wikidata: Q128395030
  wikipedia: Kotzig's conjecture
inventory: [MHFDYGVG6V]
---

## Statement

**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 $P_k$-graph if each pair of distinct vertices is connected by *exactly one* path of length $k$. Kotzig's conjecture asserts that for $k\ge 3$ there are no finite $P_k$-graphs with two or more vertices. The conjecture was first formulated by Anton Kotzig in 1974. It has been verified for $k\le 20$, but remains open in the general case (as of July 2026).