Max-Cut in Degenerate H-Free Graphs
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We obtain several lower bounds on the Max-Cut of d-degenerate H-free graphs. Let f(m,d,H) denote the smallest Max-Cut of an H-free d-degenerate graph on m edges. We show that f(m,d,K_r)>(1/2 + d^-1+Ω(r^-1))m, improving on and generalizing a recent work of Carlson, Kolla, and Trevisan. We also give bounds on f(m,d,H) when H is a cycle, odd wheel, or a complete bipartite graph with at most 4 vertices on one side. We also show stronger bounds on f(m,d,K_r) assuming a conjecture of Alon, Bollabas, Krivelevich, and Sudakov (2003). We conjecture that f(m,d,K_r)= ( 1/2 + Θ_r(d^-1/2) )m for every r> 3, and show that this conjecture implies the ABKS conjecture.
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