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
Graphs and Combinatorics – Springer Journals
Published: Jun 6, 2018
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