arXiv subjects

Michael Young

Publications and source records attributed to Michael Young.

46 records · Page 3Linked to original sources

arXiv2014

Rainbow triangles in three-colored graphs

Erdos and Sos proposed a problem of determining the maximum number F(n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that F(n) = F(a)+ F(b)+F(c)+F(d)+abc+abd+acd+bcd, where a+b+c+d = n and a, b, c, d are as equal as possible. We prove that the conjectured recurrence holds for sufficiently large n. We also prove the conjecture for n = 4k for all k. These results imply that lim F(n) n^3/6 = 0.4, and determine the unique limit object. In the proof we use flag algebras combined with stability arguments.

math.CO↗

arXiv2013

On Weak Chromatic Polynomials of Mixed Graphs

A \emph{mixed graph} is a graph with directed edges, called arcs, and undirected edges. A $k$-coloring of the vertices is proper if colors from ${1,2,...,k}$ are assigned to each vertex such that $u$ and $v$ have different colors if $uv$ is an edge, and the color of $u$ is less than or equal to (resp. strictly less than) the color of $v$ if $uv$ is an arc. The weak (resp. strong) chromatic polynomial of a mixed graph counts the number of proper $k$-colorings. Using order polynomials of partially ordered sets, we establish a reciprocity theorem for weak chromatic polynomials giving interpretations of evaluations at negative integers.

math.CO↗

arXiv2013

Nowhere-Zero $\vec k$-Flows on Graphs

We introduce and study a multivariate function that counts nowhere-zero flows on a graph G, in which each edge of G has an individual capacity. We prove that the associated counting function is a piecewise-defined polynomial in these capacities, which satisfy a combinatorial reciprocity law that incorporates totally cyclic orientations of G.

math.CO↗

arXiv2013

Sum list coloring, the sum choice number, and sc-greedy graphs

Let G=(V,E) be a graph and let f be a function that assigns list sizes to the vertices of G. It is said that G is f-choosable if for every assignment of lists of colors to the vertices of G for which the list sizes agree with f, there exists a proper coloring of G from the lists. The sum choice number is the minimum of the sum of list sizes for f over all choosable functions f for G. The sum choice number of a graph is always at most the sum |V|+|E|. When the sum choice number of G is equal to this upper bound, G is said to be sc-greedy. In this paper, we determine the sum choice number of all graphs on five vertices, show that trees of cycles are sc-greedy, and present some new general results about sum list coloring.

math.CO↗

arXiv2013

Large-Scale Learning with Less RAM via Randomization

We reduce the memory footprint of popular large-scale online learning methods by projecting our weight vector onto a coarse discrete set using randomized rounding. Compared to standard 32-bit float encodings, this reduces RAM usage by more than 50% during training and by up to 95% when making predictions from a fixed model, with almost no loss in accuracy. We also show that randomized counting can be used to implement per-coordinate learning rates, improving model quality with little additional RAM. We prove these memory-saving methods achieve regret guarantees similar to their exact variants. Empirical evaluation confirms excellent performance, dominating standard approaches across memory versus accuracy tradeoffs.

cs.LG↗

arXiv2012

On diamond-free subposets of the Boolean lattice

The Boolean lattice of dimension two, also known as the diamond, consists of four distinct elements with the following property: $A\subset B,C\subset D$. A diamond-free family in the $n$-dimensional Boolean lattice is a subposet such that no four elements form a diamond. Note that elements $B$ and $C$ may or may not be related. There is a diamond-free family in the $n$-dimensional Boolean lattice of size $(2-o(1)){n\choose\lfloor n/2\rfloor}$. In this paper, we prove that any diamond-free family in the $n$-dimensional Boolean lattice has size at most $(2.25+o(1)){n\choose\lfloor n/2\rfloor}$. Furthermore, we show that the so-called Lubell function of a diamond-free family in the $n$-dimensional Boolean lattice is at most $2.25+o(1)$, which is asymptotically best possible.

math.CO↗

arXiv2012

Monochromatic triangles in three-coloured graphs

In 1959, Goodman determined the minimum number of monochromatic triangles in a complete graph whose edge set is two-coloured. Goodman also raised the question of proving analogous results for complete graphs whose edge sets are coloured with more than two colours. In this paper, we determine the minimum number of monochromatic triangles and the colourings which achieve this minimum in a sufficiently large three-coloured complete graph.

math.CO↗

arXiv2011

Logic circuits from zero forcing

We design logic circuits based on the notion of zero forcing on graphs; each gate of the circuits is a gadget in which zero forcing is performed. We show that such circuits can evaluate every monotone Boolean function. By using two vertices to encode each logical bit, we obtain universal computation. We also highlight a phenomenon of "back forcing" as a property of each function. Such a phenomenon occurs in a circuit when the input of gates which have been already used at a given time step is further modified by a computation actually performed at a later stage. Finally, we point out that zero forcing can be also used to implement reversible computation. The model introduced here provides a potentially new tool in the analysis of Boolean functions, with particular attention to monotonicity.

cs.DM↗

arXiv2011

Zero forcing, linear and quantum controllability for systems evolving on networks

We study the dynamics of systems on networks from a linear algebraic perspective. The control theoretic concept of controllability describes the set of states that can be reached for these systems. Under appropriate conditions, there is a connection between the quantum (Lie theoretic) property of controllability and the linear systems (Kalman) controllability condition. We investigate how the graph theoretic concept of a zero forcing set impacts the controllability property. In particular, we prove that if a set of vertices is a zero forcing set, the associated dynamical system is controllable. The results open up the possibility of further exploiting the analogy between networks, linear control systems theory, and quantum systems Lie algebraic theory. This study is motivated by several quantum systems currently under study, including continuous quantum walks modeling transport phenomena. Additionally, it proposes zero forcing as a new notion in the analysis of complex networks.

quant-ph↗

arXiv2007

Nanofilters for Optical Nanocircuits

We theoretically and numerically study the design of optical 'lumped' nanofiltering devices in the framework of our recently proposed paradigm for optical nanocircuits. In particular, we present the design of basic filtering elements, such as low-pass, band-pass, stop-band and high-pass 'lumped' nanofilters, for use in optical nanocircuits together with more complex designs, such as multi-zero or multi-pole nanofilters, to work at THz, infrared and optical frequencies. Following the nanocircuit theory, we show how it is possible to design such complex frequency responses by simple rules, similar to RF circuit design, and we compare the frequency response of these optical nanofilters with classic filters in RF circuits. These results may provide a theoretical foundation for fabricating nanofilters in optical lumped nanocircuit devices.

cond-mat.mtrl-sci↗