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

Cliques in the Union of \$\$C_4\$\$ C 4 -Free Graphs Graphs and Combinatorics https://doi.org/10.1007/s00373-018-1898-4 ORIGINAL PAPER Cliques in the Union of C -Free Graphs 1 1 Abeer Othman · Eli Berger Received: 18 July 2016 / Revised: 13 March 2018 © Springer Japan KK, part of Springer Nature 2018 Abstract Let B and R be two simple C -free graphs with the same vertex set V , and let B ∨ R be the simple graph with vertex set V and edge set E (B) ∪ E (R). We prove that if B ∨ R is a complete graph, then there exists a B-clique X,an R-clique Y and aset Z which is a clique both in B and in R, such that V = X ∪ Y ∪ Z. For general B and R, not necessarily forming together a complete graph, we obtain that ω(B ∨ R) ≤ ω(B) + ω(R) + min(ω(B), ω(R)) and ω(B ∨ R) ≤ ω(B) + ω(R) + ω(B ∧ R) where B ∧ R is the simple graph with vertex set V and edge set E (B) ∩ E (R). Keywords C -free graphs · Cliques · Obedient sets This research is partially supported by the United States—Israel Binational Science Foundation Grants 2012031 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 OnlineFirst – Jun 6, 2018
6 pages

/lp/springer_journal/cliques-in-the-union-of-c-4-c-4-free-graphs-80ioX5LL32
Publisher
Springer Japan
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

Graphs and Combinatorics https://doi.org/10.1007/s00373-018-1898-4 ORIGINAL PAPER Cliques in the Union of C -Free Graphs 1 1 Abeer Othman · Eli Berger Received: 18 July 2016 / Revised: 13 March 2018 © Springer Japan KK, part of Springer Nature 2018 Abstract Let B and R be two simple C -free graphs with the same vertex set V , and let B ∨ R be the simple graph with vertex set V and edge set E (B) ∪ E (R). We prove that if B ∨ R is a complete graph, then there exists a B-clique X,an R-clique Y and aset Z which is a clique both in B and in R, such that V = X ∪ Y ∪ Z. For general B and R, not necessarily forming together a complete graph, we obtain that ω(B ∨ R) ≤ ω(B) + ω(R) + min(ω(B), ω(R)) and ω(B ∨ R) ≤ ω(B) + ω(R) + ω(B ∧ R) where B ∧ R is the simple graph with vertex set V and edge set E (B) ∩ E (R). Keywords C -free graphs · Cliques · Obedient sets This research is partially supported by the United States—Israel Binational Science Foundation Grants 2012031

### Journal

Graphs and CombinatoricsSpringer Journals

Published: Jun 6, 2018

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