离散数学

The Additive Coloring Problem is a variation of the Coloring Problem where labels of $\{1,\ldots,k\}$ are assigned to the vertices of a graph $G$ so that the sum of labels over the neighborhood of each vertex is a proper coloring of $G$.…

Our goal is to prove new results in graph theory and combinatorics thanks to the speed of computers, used with smart algorithms. We tackle four problems. The four-colour theorem states that any map whose countries are connected can be…

We generalize Axel Thue's familiar definition of overlaps in words, and show that there are no infinite words containing split occurrences of these generalized overlaps. Along the way we prove a useful theorem about repeated disjoint…

We study the mixing time of random walks on small-world networks modelled as follows: starting with the 2-dimensional periodic grid, each pair of vertices $\{u,v\}$ with distance $d>1$ is added as a "long-range" edge with probability…

In this paper, we study the feedback game on $3$-chromatic Eulerian triangulations of surfaces. We prove that the winner of the game on every $3$-chromatic Eulerian triangulation of a surface all of whose vertices have degree $0$ modulo $4$…

We obtain a description of the Bipartite Perfect Matching decision problem as a multilinear polynomial over the Reals. We show that it has full degree and $(1-o_n(1))\cdot 2^{n^2}$ monomials with non-zero coefficients. In contrast, we show…

We propose the total staggered quantum walk model and the total tessellation cover of a graph. This model uses the concept of total tessellation cover to describe the motion of the walker who is allowed to hop both to vertices and edges of…

In spatial networks vertices are arranged in some space and edges may cross. When arranging vertices in a 1-dimensional lattice edges may cross when drawn above the vertex sequence as it happens in linguistic and biological networks. Here…

This paper is a gentle but rigorous introduction to quantum computing intended for discrete mathematicians. Starting from a small set of assumptions on the behavior of quantum computing devices, we analyze their main characteristics,…

We introduce the Iterated Global model as a deterministic graph process that simulates several properties of complex networks. In this model, for every set $S$ of nodes of a prescribed cardinality, we add a new node that is adjacent to…

In 1979, Nishizeki and Baybars showed that every planar graph with minimum degree 3 has a matching of size $\frac{n}{3}+c$ (where the constant $c$ depends on the connectivity), and even better bounds hold for planar graphs with minimum…

Past work on evacuation planning assumes that evacuees will follow instructions -- however, there is ample evidence that this is not the case. While some people will follow instructions, others will follow their own desires. In this paper,…

In the list coloring problem for two matroids, we are given matroids $M_1=(S,{\cal I}_1)$ and $M_2=(S,{\cal I}_2)$ on the same ground set $S$, and the goal is to determine the smallest number $k$ such that given arbitrary lists $L_s$ of $k$…

We consider extension of a closure system on a finite set S as a closure system on the same set S containing the given one as a sublattice. A closure system can be represented in different ways, e.g. by an implicational base or by the set…

The bigraph theory is a relatively young, yet formally rigorous, mathematical framework encompassing Robin Milner's previous work on process calculi, on the one hand, and provides a generic meta-model for complex systems such as multi-agent…

Given a relational database, a key is a set of attributes such that a value assignment to this set uniquely determines the values of all other attributes. The database uniquely defines a pure Horn function $h$, representing the functional…

Given a CNF formula $\Phi$ with clauses $C_1,\ldots,C_m$ and variables $V=\{x_1,\ldots,x_n\}$, a truth assignment $a:V\rightarrow\{0,1\}$ of $\Phi$ leads to a clause sequence $\sigma_\Phi(a)=(C_1(a),\ldots,C_m(a))\in\{0,1\}^m$ where $C_i(a)…

How might one "reduce" a graph? That is, generate a smaller graph that preserves the global structure at the expense of discarding local details? There has been extensive work on both graph sparsification (removing edges) and graph…

The problem "Given a Boolean function $f$ of $n$ variables by its truth table vector. Find (if exists) a vector $\alpha \in \{0,1\}^n$ of maximal (or minimal) weight, such that $f(\alpha)= 1$." is considered here. It is closely related to…

A dominating set $S$ is an Isolate Dominating Set (IDS) if the induced subgraph $G[S]$ has at least one isolated vertex. In this paper, we initiate the study of new domination parameter called, isolate secure domination. An isolate…