Embeddings of graphs in euclidean spaces

AbstractThe 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! $$\mathop x\limits^ - \mathop y\limits^ - \geqslant t$$ . 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 inR6. We prove that d(G)≥1/15n for most graphs and that there are 3-regular graphs with d(G)≥c logn/log logn.