For　one-dimensional　nonlinear　lattices　with　momentum　conserving　interparticle　interactions,　intensive　studies　have　suggested　that　the　heat　conductivity　κ　diverges　with　the　system　size　L　as　κ∼Lα　and　the　value　of　α　is　universal.　But　in　the　Fermi-Pasta-Ulam-β　lattices　with　nearest-neighbor　(NN)　and　next-nearest-neighbor　(NNN)　coupling,　we　find　that　α　strongly　depends　on　γ,　the　ratio　of　the　NNN　coupling　to　the　NN　coupling.　The　correlation　between　the　γ-dependent　heat　conduction　behavior　and　the　in-band　discrete　breathers　is　also　analyzed.　©　2012　American　Physical　Society.