# Cliques in the Union of $$C_4$$ C 4 -Free Graphs

Cliques in the Union of $$C_4$$ C 4 -Free Graphs Let B and R be two simple $$C_4$$ C 4 -free graphs with the same vertex set V, and let $$B \vee R$$ B ∨ R be the simple graph with vertex set V and edge set $$E(B) \cup E(R)$$ E ( B ) ∪ E ( R ) . We prove that if $$B \vee R$$ B ∨ R is a complete graph, then there exists a B-clique X, an R-clique Y and a set Z which is a clique both in B and in R, such that $$V=X\cup Y\cup Z$$ V = X ∪ Y ∪ Z . For general B and R, not necessarily forming together a complete graph, we obtain that \begin{aligned} \omega (B \vee R)\le & {} \omega (B)+\omega (R)+\frac{1}{2}\min (\omega (B),\omega (R))\\&\hbox {and}\\ \omega (B \vee R)\le & {} \omega (B)+\omega (R)+\omega (B \wedge R ) \end{aligned} ω ( B ∨ R ) ≤ ω ( B ) + ω ( R ) + 1 2 min ( ω ( B ) , ω ( R ) ) and ω ( B ∨ R ) ≤ ω ( B ) + ω ( R ) + ω ( B ∧ R ) where $$B \wedge R$$ B ∧ R is the simple graph with vertex set V and edge set $$E(B) \cap E(R)$$ E ( B ) ∩ E ( R ) . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Graphs and Combinatorics Springer Journals

# Cliques in the Union of $$C_4$$ C 4 -Free Graphs

, Volume 34 (4) – Jun 6, 2018
6 pages

/lp/springer_journal/cliques-in-the-union-of-c-4-c-4-free-graphs-80ioX5LL32
Publisher
Springer Journals
Subject
Mathematics; Combinatorics; Engineering Design
ISSN
0911-0119
eISSN
1435-5914
D.O.I.
10.1007/s00373-018-1898-4
Publisher site
See Article on Publisher Site

### Abstract

Let B and R be two simple $$C_4$$ C 4 -free graphs with the same vertex set V, and let $$B \vee R$$ B ∨ R be the simple graph with vertex set V and edge set $$E(B) \cup E(R)$$ E ( B ) ∪ E ( R ) . We prove that if $$B \vee R$$ B ∨ R is a complete graph, then there exists a B-clique X, an R-clique Y and a set Z which is a clique both in B and in R, such that $$V=X\cup Y\cup Z$$ V = X ∪ Y ∪ Z . For general B and R, not necessarily forming together a complete graph, we obtain that \begin{aligned} \omega (B \vee R)\le & {} \omega (B)+\omega (R)+\frac{1}{2}\min (\omega (B),\omega (R))\\&\hbox {and}\\ \omega (B \vee R)\le & {} \omega (B)+\omega (R)+\omega (B \wedge R ) \end{aligned} ω ( B ∨ R ) ≤ ω ( B ) + ω ( R ) + 1 2 min ( ω ( B ) , ω ( R ) ) and ω ( B ∨ R ) ≤ ω ( B ) + ω ( R ) + ω ( B ∧ R ) where $$B \wedge R$$ B ∧ R is the simple graph with vertex set V and edge set $$E(B) \cap E(R)$$ E ( B ) ∩ E ( R ) .

### Journal

Graphs and CombinatoricsSpringer Journals

Published: Jun 6, 2018

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