# The Complexity of HCP in Digraps with Degree Bound Two

## TL;DR

This paper demonstrates that the Hamiltonian cycle problem in directed graphs with maximum degree two can be solved in polynomial time using novel mappings, challenging the typical NP-complete classification.

## Contribution

It introduces two mappings that establish polynomial solutions for HCP in degree-two digraphs and claims to prove P=NP based on these results.

## Key findings

- HCP in degree-two digraphs is polynomially solvable.
- A bijection between incidence matrices of digraphs and bipartite graphs.
- Finding non-isomorphic Hamiltonian cycles is also polynomial.

## Abstract

The Hamiltonian cycle problem (HCP) in digraphs D with degree bound two is solved by two mappings in this paper. The first bijection is between an incidence matrix C_{nm} of simple digraph and an incidence matrix F of balanced bipartite undirected graph G; The second mapping is from a perfect matching of G to a cycle of D. It proves that the complexity of HCP in D is polynomial, and finding a second non-isomorphism Hamiltonian cycle from a given Hamiltonian digraph with degree bound two is also polynomial. Lastly it deduces P=NP base on the results.