Clique Numbers of Graphs and Irreducible Exact m-Covers of Z

TL;DR

This paper constructs specific graphs with prescribed clique numbers and demonstrates the existence of certain exact m-covers of integers that cannot be decomposed into simpler covers, advancing understanding in graph theory and number covers.

Contribution

It introduces a method to construct graphs with controlled clique numbers and proves the existence of complex exact m-covers of Z that are not unions of two 1-covers.

Findings

Constructed graphs with clique number m and arbitrary partitions maintaining clique number.

Proved existence of exact m-covers of Z that are not unions of two 1-covers.

Abstract

For each m>=1 and k>=2, we construct a graph G=(V,E) with \omega(G)=m such that max_{1\leq i\leq k} \omega(G[V_i])=m for arbitrary partition V=V_1\cup...\cup V_k, where \omega(G) is the clique number of G and G[V_i] is the induced subgraph of G with the vertex set V_i. Using this result, we show that for each m>=2 there exists an exact m-cover of Z which is not the union of two 1-covers.