Vertex 2-coloring without monochromatic cycles

TL;DR

This paper investigates the computational complexity of two-coloring vertices in graphs to avoid monochromatic cycles of specific lengths, proving the problem is computationally hard through a reduction from SAT.

Contribution

It introduces a novel reduction from SAT to the vertex 2-coloring problem avoiding monochromatic cycles, highlighting the problem's computational difficulty.

Findings

The problem is NP-hard for certain cycle lengths.

Properties of 2-coloring cliques are crucial for the reduction.

The reduction demonstrates the problem's computational intractability.

Abstract

In this paper we study a problem of vertex two-coloring of undirected graph such that there is no monochromatic cycle of given length. We show that this problem is hard to solve. We give a proof by presenting a reduction from variation of satisfiability (SAT) problem. We show nice properties of coloring cliques with two colors which plays pivotal role in the reduction construction.