Centre for Discrete and Applicable Mathematics 

CDAM Research Report, LSECDAM200701January 2007 
Cutting Two Graphs Simultaneously
Viresh Patel
Consider two graphs, G_{1} and G_{2}, 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) ≥ m_{1}/2 and e_{G2}(A,B)≥ m_{2}/2Δ(G_{2})/2. We also show, using a probabilistic method, that if G_{1} and G_{2} belong to certain classes of graphs, (for instance, if G_{1} and G_{2} both have a density of at least 2/3, or if G_{1} and G_{2} 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_{Gi}(A,B)≥ m_{i}/2 for i = 1, 2.A PDF file (162 kB) with the full contents of this report can be downloaded by clicking here.
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