Centre for Discrete and Applicable Mathematics

CDAM Research Report, LSE-CDAM-2007-01

January 2007

Cutting Two Graphs Simultaneously

Viresh Patel

Consider two graphs, G₁ and G₂, on the same vertex set V, with |V| = n and G_i having m_i edges for i = 1, 2. We give a simple algorithm that partitions V into sets A and B such that e_G₁(A,B) ≥ m₁/2 and e_G₂(A,B)≥ m₂/2-Δ(G₂)/2. We also show, using a probabilistic method, that if G₁ and G₂ belong to certain classes of graphs, (for instance, if G₁ and G₂ both have a density of at least 2/3, or if G₁ and G₂ are both regular of degree at most (n/16) - 6 with n sufficiently large) then we can find a partition of V into sets A and B such that e_{G_i}(A,B)≥ m_i/2 for i = 1, 2.

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		CDAM Research Reports Series Centre for Discrete and Applicable Mathematics London School of Economics Houghton Street London WC2A 2AE, U.K.
		Phone: +44(0)-20-7955 7494. Fax: +44(0)-20-7955 6877. Email: info@maths.lse.ac.uk