Clique numbers of graph unions
Maria Chudnovsky, Juba Ziani

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 and be two simple graphs with vertex set , and let be the simple graph with vertex set , in which two vertices are adjacent if they are adjacent in at least one of and . For , we denote by the subgraph of induced by ; let and be defined similarly. We say that the pair is {\em additive} if for every , the sum of the clique numbers of and is at least the clique number of . 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}.
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Taxonomy
TopicsGraph theory and applications · Advanced Graph Theory Research · Limits and Structures in Graph Theory
