Searcharxiv⌕ Search

arXiv subjects

Le Anh Vinh

Publications and source records attributed to Le Anh Vinh.

47 records · Page 3Linked to original sources
arXiv2008

Explicit tough Ramsey graphs

A graph G is t-tough if any induced subgraph of it with x > 1 connected components is obtained from G by deleting at least tx vertices. Chvatal conjectured that there exists an absolute constant t_0 so that every t_0-tough graph is pancyclic. This conjecture was disproved by Bauer, van den Heuvel and Schmeichel by constructing a t_0-tough triangle-free graph for every real t_0. For each finite field F_q with q odd, we consider graphs associated to the finite Euclidean plane and the finite upper half plane over F_q. These graphs have received serious attention as they have been shown to be Ramanujan (or asymptotically Ramanujan) for large q. We will show that for infinitely many q, these graphs provide further counterexamples to Chvatal's conjecture. They also provide a good constructive lower bound for the Ramsey number R(3,k).

math.CO↗
arXiv2008

On a Furstenberg-Katznelson-Weiss type theorem over finite fields

Using Fourier analysis, Covert, Hart, Iosevich and Uriarte-Tuero (2008) showed that if the cardinality of a subset of the 2-dimensional vector space over a finite field with q elements is >= rq^2, with q^{-1/2} << r <= 1 then it contains an isometric copy of >= crq^3 triangles. In this note, we give a graph theoretic proof of this result.

math.CO↗
arXiv2007

Explicit Ramsey graphs and Erdos distance problem over finite Euclidean and non-Euclidean spaces

We study the Erdos distance problem over finite Euclidean and non-Euclidean spaces. Our main tools are graphs associated to finite Euclidean and non-Euclidean spaces that are considered in Bannai-Shimabukuro-Tanaka (2004, 2007). These graphs are shown to be asymptotically Ramanujan graphs. The advantage of using these graphs is twofold. First, we can derive new lower bounds on the Erdos distance problems with explicit constants. Second, we can construct many explicit tough Ramsey graphs R(3,k).

math.CO↗
arXiv2006

Some colouring problems for unit-quadrance graphs

The quadrance between two points $A_1 = (x_1, y_1)$ and $A_2 = (x_2, y_2)$ is the number $Q (A_1, A_2) = (x_1 - x_2)^2 + (y_1 - y_2)^2$. Let $q$ be an odd prime power and $F_q$ be the finite field with $q$ elements. The unit-quadrance graph $D_q$ has the vertex set $F_q^2$, and $X, Y \in F_q^2$ are adjacent if and only if $Q (A_1, A_2) = 1$. In this paper, we study some colouring problems for the unit-quadrance graph $D_q$ and discuss some open problems.

math.CO↗
arXiv2006

Divisor graphs have arbitrary order and size

A divisor graph $G$ is an ordered pair $(V, E)$ where $V \subset \mathbbm{Z}$ and for all $u \neq v \in V$, $u v \in E$ if and only if $u \mid v$ or $v \mid u$. A graph which is isomorphic to a divisor graph is also called a divisor graph. In this note, we will prove that for any $n \geqslant 1$ and $0 \leqslant m \leqslant \binom{n}{2}$ then there exists a divisor graph of order $n$ and size $m$. We also present a simple proof of the characterization of divisor graphs which is due to Chartran, Muntean, Saenpholpant and Zhang.

math.CO↗
arXiv2006

Random walks on hypergroup of conics in finite fields

In this paper we study random walks on the hypergroup of conics in finite fields. We investigate the behavior of random walks on this hypergroup, the equilibrium distribution and the mixing times. We use the coupling method to show that the mixing time of random walks on hypergroup of conics is only linear.

math.PR↗
arXiv2005

Two identities of derangements

In this note, we present two new identities for derangements. As a corollary, we have a combinatorial proof of the irreducibility of the standard representation of symmetric groups.

math.CO↗
arXiv2005

On chromatic number of unit-quadrance graphs (finite Euclidean graphs)

The quadrance between two points A_1=(x_1, y_1) and A_2=(x_2, y_2) is the number Q (A_1, A_2) = (x_1 - x_2)^2 + (y_1 - y_2)^2. Let q be an odd prime power and F_q be the finite field with $q$ elements. The unit-quadrance graph D_q has the vertex set F_q^2, and X, Y in F_q^2 are adjacent if and only if Q(A_1, A_2) = 1. Let χ(F_q^2) be the chromatic number of graph D_q. In this note, we will show that q^{1/2}(1/2+o(1)) <= χ(F_q^2) <= q(1/2 + o(1)). As a corollary, we have a construction of triangle-free graphs D_q of order q^2 with χ(D_q) >= q/2 for infinitely many values of q.

math.CO↗
arXiv2005

Quadrance polygons, association schemes and strongly regular graphs

Quadrance between two points A_1 = [x_1,y_1] and A_2 = [x_2,y_2] is the number Q (A_1, A_2) := (x_2 - x_1)^2 + (y_2 - y_1)^2. In this paper, we present some interesting results arise from this notation. In Section 1, we will study geometry over finite fields under quadrance notations. The main purpose of Section 1 is to answer the question, for which a_1,...,a_n, we have a polygon A_1...A_n such that Q(A_i,A_{i+1})=a_i for i = 1,...,n. In Section 2, using tools developed in Section 1, we define a family of association schemes over finite field space F_q x F_q where q is a prime power. These schemes give rise to a graph V_q with vertices the points of F_q^2, and where (X,Y) is an edge of V_q if and only if Q(X,Y) is a nonzero square number in F_q. In Section 3, we will show that V_q is a strongly regular graph and propose a conjecture about the maximal clique of V_q.

math.CO↗
arXiv2005

Ramdom walks on hypergroup of circles in finite fields

In this paper we study random walks on the hypergroup of circles in a finite field of prime order p = 4l + 3. We investigating the behavior of random walks on this hypergroup, the equilibrium distribution and the mixing times. We use two different approaches - comparision of Dirichlet forms (geometric bound of eigenvalues), and coupling methods, to show that the mixing time of random walks on hypergroup of circles is only linear.

math.CO↗