Hamiltonian Cycles in Signed and Multisigned Complete Graphs

TL;DR

This paper characterizes the conditions under which signed and multisigned complete graphs contain Hamiltonian cycles of different multisigns, linking these properties to the signs of triangles within the graphs.

Contribution

It extends known results on signed graphs to multisigned graphs, establishing new criteria for the existence of Hamiltonian cycles with different multisigns.

Findings

Signed complete graphs contain both positive and negative Hamiltonian cycles iff they contain both positive and negative triangles.

All Hamiltonian cycles are negative iff all triangles are negative and n is odd.

All Hamiltonian cycles are positive iff all triangles are positive or all triangles are negative with n even.

Abstract

A signed complete graph contains both positive and negative Hamiltonian cycles if and only if it also contains both positive and negative triangles. Otherwise, all Hamiltonian cycles are negative if and only if all triangles are negative and n is odd, while all Hamiltonian cycles are positive if and only if all triangles are negative and n is even, or all triangles are positive. Extending these results to multisigned complete graphs, we prove that such a graph contains at least two Hamiltonian cycles with different multisigns if and only if it contains at least two triangles with different multisigns.