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]

Systematic Topology Analysis and Generation Using Degree Correlations

We present a new, systematic approach for analyzing network topologies. We first introduce the dK-series of probability distributions specifying all degree correlations within d-sized subgraphs of a given graph G. Increasing values of d capture progressively more properties of G at the cost of more complex representation of the probability distribution. Using this series, we can quantitatively measure the distance between two graphs and construct random graphs that accurately reproduce virtually all metrics proposed in the literature. The nature of the dK-series implies that it will also capture any future metrics that may be proposed. Using our approach, we construct graphs for d=0,1,2,3 and demonstrate that these graphs reproduce, with increasing accuracy, important properties of measured and modeled Internet topologies. We find that the d=2 case is sufficient for most practical purposes, while d=3 essentially reconstructs the Internet AS- and router-level topologies exactly. We hope that a systematic method to analyze and synthesize topologies offers a significant improvement to the set of tools available to network topology and protocol researchers.Comment: Final versio

Coloring Powers and Girth

International audienceAlon and Mohar (2002) posed the following problem: among all graphs G of maximum degree at most d and girth at least g, what is the largest possible value of χ(G t), the chromatic number of the tth power of G? For t ≥ 3, we provide several upper and lower bounds concerning this problem, all of which are sharp up to a constant factor as d → ∞. The upper bounds rely in part on the probabilistic method, while the lower bounds are various direct constructions whose building blocks are incidence structures. 1. Introduction. For a positive integer t, the tth power G t of a (simple) graph G = (V, E) is a graph with vertex set V in which two distinct elements of V are joined by an edge if there is a path in G of length at most t between them. What is the largest possible value of the chromatic number χ(G t) of G t , among all graphs G with maximum degree at most d and girth (the length of the shortest cycle contained in the graph) at least g? For t = 1, this question was essentially a long-standing problem of Vizing [11], one that stimulated much work on the chromatic number of bounded degree triangle-free graphs, and was eventually settled asymptotically by Johansson [6] using the probabilistic method. In particular, he showed that the largest possible value of the chromatic number over all girth 4 graphs of maximum degree at most d is Θ(d/ log d) as d → ∞. The case t = 2 was considered and settled asymptotically by Alon and Mohar [2]. They showed that the largest possible value of the chromatic number of a graph's square taken over all girth 7 graphs of maximum degree at most d is Θ(d 2 / log d) as d → ∞. Moreover, there exist girth 6 graphs of arbitrarily large maximum degree d such that the chromatic number of their square is (1 + o(1))d 2 as d → ∞. In this work, we consider this extremal question for larger powers t ≥ 3, which was posed as a problem in [2], and settle a range of cases for g. A first basic remark to make is that, ignoring the girth constraint, the maximum degree Δ(G t) of G t for G a graph of maximum degree at most d satisfie

Vortex development for a 65 degree swept back delta wing with varying thickness and maximum thickness location

Low signature, unmanned combat aerial vehicles (UCAV) are foreseen to play an important role in future military missions. Most UCAV concepts feature some sort of delta or lambda type of wing with high or moderately swept leading edge, where the fuselage is part of the lifting surface, thus, forming a blended wing body. These configurations have advantageous characteristics at high speeds and a low radar signature, making them desirable shapes for military purposes. However, blended wing bodies are known for having a serious problem with their longitudinal static stability and their performance at low speeds. Initial wing design or the ability to change the wing shape in flight, by means of wing morphing, could be utilised to overcome these issues. Therefore, with the technology readiness level of morphing configurations increasing, an understanding of the effect of slight changes to the aerofoil design need to be more thoroughly established. Therefore, to understand the change in aerodynamics when altering the thickness/chord ratio (3.4%, 6% and 12%), spanwise thickness distribution and maximum thickness location/chord (30% and 50%) a computational investigation of a 65° delta wing with different profiles of different thicknesses was undertaken. Here, the maximum thickness location was investigated on two biconvex wings with maximum thickness locations at 30% and 50% root chord, whilst a tip taper study was conducted using the NATO’s AVT-113 VFE-2 configuration and a derivation of it. The effect of thickness was investigated on all configurations. Particular emphasis was placed on the longitudinal stability and lift to drag (L/D) ratio as they both impact aircraft design. The former due to its impact on the positioning and size of control surfaces and the latter due to its impact on thrust, maximum range and maximum take-off weight considerations. The investigation was carried out at Mach number 0.1 and Reynolds number of 750,000 with the former being representative for take-off and landing conditions for some military aircraft incorporating slender wing flows. The numerical results were validated for selected configurations in the University of the West of England’s wind tunnel. For the numerical simulations ANSYS FLUENT was used with an unstructured hybrid mesh approach. The steady state runs were conducted using the k-ω Shear Stress turbulence (SST) model with curvature and low Reynolds number corrections.The study showed that tip taper, thickness and maximum thickness location affect the L/D ratio and longitudinal stability. For the biconvex wings an increase in thickness was found to increase the L/D at higher angles of attack whilst the maximum attainable L/D ratio was found to be a function of both, thickness and maximum thickness location. For wings with mainly flat upper and lower surface L/D decreased with increase in thickness and so did the maximum L/D ratio. This was irrespective of tip taper. The angle of attack at which the maximum L/D was reached moved to a slightly higher angle (α=1°) when thickness was increased on the biconvex configurations, whereby the tapered VFE-2 configuration experienced a shift of up to α= 3°. Having a constant thickness distribution further enhanced this effect moving the angle of maximum L/D from α= 5° to α= 15°. Moving the maximum thickness location forward resulted in an overall increase in L/D for the 6% and 12% configuration whilst not having a significant effect for thin wings, with the same being true for the maximum attainable L/D ratio. The angle of attack at which maximum L/D was reached moved to a lower angle by 1° for the 3.4% and 12% wing whilst being unaffected for the 6% wing. Tip taper showed to improve L/D and maximum L/D irrespective of thickness whilst the angle of attack at which maximum L/D could be achieved moved to lower angles when tip taper was introduced. This effect showed to enhance with increase in thickness.The stability was evaluated at a lift coefficient C_L=0.5 typical for take-off and landing. It was found that stability improved with increase in thickness indicated by a rearward movement of the aerodynamic centre. These improvements were more significant when maximum thickness location was moved forward and resulted in up to 1.82% increase in static margin. Similar observations were made for the VFE-2 configuration and its adapted model with span taper. Here, the static margin increased by 3.21% when introducing tip taper. This was due to increased rear loading. In particular moving the maximum thickness location forward resulted in a delay in vortex onset, thus, further improving stability. Increase in thickness also resulted in enhanced stability (by up to 6.68%) but was strongly dependent on maximum thickness location and spanwise thickness distribution.The flow pattern was also affected by increasing thickness, resulting in primary vortex stretching and domination of separation bubbles especially close to the apex. On the biconvex profiles shifting the maximum thickness location rearward helped in maintaining non-linear lift generation even when thickness was increased. It further resulted in delayed onset of vortex breakdown. Increase in thickness on wings with a flat profile was found to cause the formation of an inner co-rotating vortex similar to that observed for delta wings with round leading edges.The main finding of the study was the demonstration that by alteration of the upper surface, leading edge suction can be recovered despite having a sharp leading edge, a phenomenon normally attributed to wings with round leading edges. This, in combination with the finding that the vortex flow pattern can be altered by reducing it to separation bubbles is not only valuable for future delta wing design but also for the implementation of wing morphing. Here, depending on the flight condition, wing shape could be altered by moving the maximum thickness location or introducing tip taper to get the desired flight performance

Linear Choosability of Sparse Graphs

We study the linear list chromatic number, denoted \lcl(G), of sparse graphs. The maximum average degree of a graph $G$, denoted \mad(G), is the maximum of the average degrees of all subgraphs of $G$. It is clear that any graph $G$ with maximum degree $\Delta(G)$ satisfies \lcl(G)\ge \ceil{\Delta(G)/2}+1. In this paper, we prove the following results: (1) if \mad(G)<12/5 and $\Delta(G)\ge 3$, then \lcl(G)=\ceil{\Delta(G)/2}+1, and we give an infinite family of examples to show that this result is best possible; (2) if \mad(G)<3 and $\Delta(G)\ge 9$, then \lcl(G)\le\ceil{\Delta(G)/2}+2, and we give an infinite family of examples to show that the bound on \mad(G) cannot be increased in general; (3) if $G$ is planar and has girth at least 5, then \lcl(G)\le\ceil{\Delta(G)/2}+4.Comment: 12 pages, 2 figure

Binary linear codes with near-extremal maximum distance

Let C denote a binary linear code with length n all of whose coordinates are essential, i.e., for each coordinate there is a codeword that is not zero in that position. Then the maximum distance D is strictly bigger than n/2, and the extremum D=(n+1)/2 is attained exactly by punctured Hadamard codes. In this paper, we classify binary linear codes with D=n/2+1. All of these codes can be produced from punctured Hadamard codes in one of essentially three different ways, each having a transparent description