# Haj\'os-like theorem for signed graphs

## TL;DR

This paper introduces five graph operations to characterize signed graphs with a given chromatic number, providing a signed analogue of Hajós' theorem that highlights the foundational role of all-positive complete graphs.

## Contribution

It establishes a signed version of Hajós' theorem by showing all such graphs can be constructed from all-positive complete graphs using specific operations.

## Key findings

- Every signed graph with chromatic number q can be generated from (K_q,+)
- The five operations preserve the chromatic number
- Signed graphs are structurally linked to all-positive complete graphs

## Abstract

The paper designs five graph operations, and proves that every signed graph with chromatic number $q$ can be obtained from all-positive complete graphs $(K_q,+)$ by repeatedly applying these operations. This result gives a signed version of the Haj\'os theorem, emphasizing the role of all-positive complete graphs played in the class of signed graphs, as played in the class of unsigned graphs.