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- W1968034334 abstract "The dimension of a graphG=(V, E) is the minimum numberd such that there exists a representation % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrpepeea0de9arVe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-dir-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4bGaey% OKH4QabmiEayaaraGaeyicI4SaamOuamaaCaaaleqabaGaamizaaaa% kiaacIcacaWG4bGaeyicI4SaamOvaiaacMcaaaa!4615! $$x to bar x in R^d (x in V)$$ and a thresholdt such thatxy εE iff % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrpepeea0de9arVe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-dir-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfGaqaai% aadIhaaSqabeaacWaGaYaaOcGHsislaaGcdaWfGaqaaiaadMhaaSqa% beaacWaGascaOcGHsislaaGccqGHLjYScaWG0baaaa!44B0! $$mathop xlimits^ - mathop ylimits^ - geqslant t$$ . We prove that d(G)≤n−x(G) and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrpepeea0de9arVe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-dir-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaacaWFKb% Gaa8hkaGqaciaa+DeacaWFPaGaeyizImQaa4NBaGGaciab9jHiTmaa% kaaabaGaa4NBaaWcbeaaaaa!4195! $$d(G) leqslant n - sqrt n $$ wheren=|V| andx(G) is the chromatic number ofG; we present upper bounds for the dimension of graphs with a large girth and we show that the complement of a forest can be represented by unit vectors inR 6. We prove that d(G)≥1/15n for most graphs and that there are 3-regular graphs with d(G)≥c logn/log logn." @default.
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- W1968034334 date "1989-08-01" @default.
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- W1968034334 title "Embeddings of graphs in euclidean spaces" @default.
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- W1968034334 doi "https://doi.org/10.1007/bf02187736" @default.
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