Nombre chromatique fractionnaire, degré maximum et maille

We prove new lower bounds on the independence ratio of graphs of maximum degree ∆ ∈ {3,4,5} and girth g ∈ {6,…,12}, notably 1/3 when (∆,g)=(4,10) and 2/7 when (∆,g)=(5,8). We establish a general upper bound on the fractional chromatic number of triangle-free graphs, which implies that deduced from the fractional version of Reed's bound for triangle-free graphs and improves it as soon as ∆ ≥ 17, matching the best asymptotic upper bound known for off-diagonal Ramsey numbers. In particular, the fractional chromatic number of a triangle-free graph of maximum degree ∆ is less than 9.916 if ∆=17, less than 22.17 if ∆=50 and less than 249.06 if ∆=1000. Focusing on smaller values of ∆, we also demonstrate that every graph of girth at least 7 and maximum degree ∆ has fractional chromatic number at most min (2∆ + 2^{k-3}+k)/k pour k ∈ ℕ. In particular, the fractional chromatic number of a graph of girth 7 and maximum degree ∆ is at most (2∆+9)/5 when ∆ ∈ [3,8], at most (∆+7)/3 when ∆ ∈  [8,20], at most (2∆+23)/7 when ∆ ∈ [20,48], and at most ∆/4+5 when ∆ ∈ [48,112]

Monochromatic Clique Decompositions of Graphs

Let $G$ be a graph whose edges are coloured with $k$ colours, and $\mathcal H=(H_1,\dots , H_k)$ be a $k$-tuple of graphs. A monochromatic $\mathcal H$-decomposition of $G$ is a partition of the edge set of $G$ such that each part is either a single edge or forms a monochromatic copy of $H_i$ in colour $i$, for some $1\le i\le k$. Let $\phi_{k}(n,\mathcal H)$ be the smallest number $\phi$, such that, for every order-$n$ graph and every $k$-edge-colouring, there is a monochromatic $\mathcal H$-decomposition with at most $\phi$ elements. Extending the previous results of Liu and Sousa ["Monochromatic $K_r$-decompositions of graphs", Journal of Graph Theory}, 76:89--100, 2014], we solve this problem when each graph in $\mathcal H$ is a clique and $n\ge n_0(\mathcal H)$ is sufficiently large.Comment: 14 pages; to appear in J Graph Theor

Bipartite induced density in triangle-free graphs

We prove that any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2/(2n)$. This is sharp up to a logarithmic factor in $n$. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of $n/d$ and $(2+o(1))\sqrt{n/\log n}$ as $n\to\infty$. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most $O(\min\{\sqrt{n},(n\log n)/d\})$ as $n\to\infty$. Relatedly, we also make two conjectures. First, any triangle-free graph on $n$ vertices has fractional chromatic number at most $(\sqrt{2}+o(1))\sqrt{n/\log n}$ as $n\to\infty$. Second, any triangle-free graph on $n$ vertices has list chromatic number at most $O(\sqrt{n/\log n})$ as $n\to\infty$.Comment: 20 pages; in v2 added note of concurrent work and one reference; in v3 added more notes of ensuing work and a result towards one of the conjectures (for list colouring

Ramsey fringes formation during excitation of topological modes in a Bose-Einstein condensate

The Ramsey fringes formation during the excitation of topological coherent modes of a Bose-Einstein condensate by an external modulating field is considered. The Ramsey fringes appear when a series of pulses of the excitation field is applied. In both Rabi and Ramsey interrogations, there is a shift of the population maximum transfer due to the strong non-linearity present in the system. It is found that the Ramsey pattern itself retains information about the accumulated relative phase between both ground and excited coherent modes.Comment: Latex file, 12 pages, 5 figure

Hyper-Ramsey Spectroscopy of Optical Clock Transitions

We present non-standard optical Ramsey schemes that use pulses individually tailored in duration, phase, and frequency to cancel spurious frequency shifts related to the excitation itself. In particular, the field shifts and their uncertainties of Ramsey fringes can be radically suppressed (by 2-4 orders of magnitude) in comparison with the usual Ramsey method (using two equal pulses) as well as with single-pulse Rabi spectroscopy. Atom interferometers and optical clocks based on two-photon transitions, heavily forbidden transitions, or magnetically induced spectroscopy could significantly benefit from this method. In the latter case these frequency shifts can be suppressed considerably below a fractional level of 10^{-17}. Moreover, our approach opens the door for the high-precision optical clocks based on direct frequency comb spectroscopy.Comment: 5 pages, 4 figure

Seconds-scale coherence in a tweezer-array optical clock

Optical clocks based on atoms and ions achieve exceptional precision and accuracy, with applications to relativistic geodesy, tests of relativity, and searches for dark matter. Achieving such performance requires balancing competing desirable features, including a high particle number, isolation of atoms from collisions, insensitivity to motional effects, and high duty-cycle operation. Here we demonstrate a new platform based on arrays of ultracold strontium atoms confined within optical tweezers that realizes a novel combination of these features by providing a scalable platform for isolated atoms that can be interrogated multiple times. With this tweezer-array clock, we achieve greater than 3 second coherence times and record duty cycles up to 96%, as well as stability commensurate with leading platforms. By using optical tweezer arrays --- a proven platform for the controlled creation of entanglement through microscopic control --- this work further promises a new path toward combining entanglement enhanced sensitivities with the most precise optical clock transitions