Clique numbers of graph unions

TL;DR

This paper characterizes when two graphs on the same vertex set have clique numbers that sum to at least the clique number of their union, providing a necessary and sufficient condition for such additive pairs.

Contribution

It offers a complete characterization of additive pairs of graphs, addressing a numerical variant of a structural graph theory problem.

Findings

Provides a necessary and sufficient condition for additive graph pairs

Addresses a numerical variant of a structural graph problem

Advances understanding of clique number behavior in graph unions

Abstract

Let $B$ and $R$ be two simple graphs with vertex set $V$, and let $G(B,R)$ be the simple graph with vertex set $V$, in which two vertices are adjacent if they are adjacent in at least one of $B$ and $R$. For $X \subseteq V$, we denote by $B|X$ the subgraph of $B$ induced by $X$; let $R|X$ and $G(B,R)|X$ be defined similarly. We say that the pair $(B,R)$ is {\em additive} if for every $X \subseteq V$, the sum of the clique numbers of $B|X$ and $R|X$ is at least the clique number of $G(B,R)|X$. In this paper we give a necessary and sufficient characterization of additive pairs of graphs. This is a numerical variant of a structural question studied in \cite{ABC}.