Prime bound of a graph

TL;DR

This paper investigates the prime bound of a graph, establishing formulas based on the size of the largest module that is a clique or stable set, and characterizes when the bound increases by one.

Contribution

It provides exact formulas for the prime bound p(G) based on the size of the largest module m(G), including special cases involving powers of two and isolated vertices.

Findings

p(G)=⌈log2(m(G))⌉ when m(G)≥2 and log2(m(G)) is not an integer

p(G)=k or k+1 when m(G)=2^k, with p(G)=k+1 iff G or its complement has 2^k isolated vertices

p(G)=1 for non-prime graphs with m(G)=1 and at least 4 vertices

Abstract

Given a graph G, a subset M of V (G) is a module of G if for each v \in V (G) \diagdownM, v is adjacent to all the elements of M or to none of them. For instance, V(G), \varnothing and {v} (v \in V(G)) are modules of G called trivial. Given a graph G, m(G) denotes the largest integer m such that there is a module M of G which is a clique or a stable set in G with |M|=m. A graph G is prime if |V(G)|\geq4 and if all its modules are trivial. The prime bound of G is the smallest integer p(G) such that there is a prime graph H with V(H)\supseteqV(G), H[V(G)] = G and |V(H)\diagdownV(G)|=p(G). We establish the following. For every graph G such that m(G)\geq2 and log_2(m(G)) is not an integer, p(G)=\lceil log_2(m(G)) \rceil. Then, we prove that for every graph G such that m(G)=2^k where k\geq1, p(G)=k or k + 1. Moreover p(G)=k+1 if and only if G or its complement admits 2^k isolated vertices. Lastly, we show that p(G) = 1 for every non-prime graph G such that |V(G)|\geq4 and m(G)=1.