离散数学

The symmetric version of Fitch's xenology relation coincides with class of complete multipartite graph and thus cannot convey any non-trivial phylogenetic information.

Many polynomial invariants are defined on graphs for encoding the combinatorial information and researching them algebraically. In this paper, we introduce the cycle polynomial and the path polynomial of directed graphs for counting cycles…

Simultaneous Embedding with Fixed Edges (SEFE) is a problem where given $k$ planar graphs we ask whether they can be simultaneously embedded so that the embedding of each graph is planar and common edges are drawn the same. Problems of SEFE…

Given a non-decreasing sequence S = (s 1,s 2,. .. ,s k) of positive integers, an S-packing edge-coloring of a graph G is a partition of the edge set of G into k subsets {X 1 ,X 2,. .. ,X k } such that for each 1 $\le$ i $\le$ k, the…

For a graph $G=(V,E)$, a set $D \subseteq V$ is called a semitotal dominating set of $G$ if $D$ is a dominating set of $G$, and every vertex in $D$ is within distance~$2$ of another vertex of~$D$. The \textsc{Minimum Semitotal Domination}…

Based on a non-rigorous formalism called the "cavity method", physicists have put forward intriguing predictions on phase transitions in discrete structures. One of the most remarkable ones is that in problems such as random $k$-SAT or…

The independence number of a sparse random graph G(n,m) of average degree d=2m/n is well-known to be \alpha(G(n,m))~2n ln(d)/d with high probability. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size (1+o(1)) n…

In this paper, we investigate Hamiltonian path problem in the context of split graphs, and produce a dichotomy result on the complexity of the problem. Our main result is a deep investigation of the structure of $K_{1,4}$-free split graphs…

A graph is $(k_1,k_2)$-colorable if its vertex set can be partitioned into a graph with maximum degree at most $k_1$ and and a graph with maximum degree at most $k_2$. We show that every $(C_3,C_4,C_6)$-free planar graph is…

We study the $q$-state ferromagnetic Potts model on the $n$-vertex complete graph known as the mean-field (Curie-Weiss) model. We analyze the Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation…

Graph isomorphism is an important computer science problem. The problem for the general case is unknown to be in polynomial time. The base algorithm for the general case works in quasi-polynomial time. The solutions in polynomial time for…

In the planted bisection model a random graph $G(n,p_+,p_- )$ with $n$ vertices is created by partitioning the vertices randomly into two classes of equal size (up to $\pm1$). Any two vertices that belong to the same class are linked by an…

The Prisoner's Dilemma Process on a graph $G$ is an iterative process where each vertex, with a fixed strategy (cooperate or defect), plays the game with each of its neighbours. At the end of a round each vertex may change its strategy to…

We introduce a Markov Chain Monte Carlo algorithm which samples from the space of spanning trees of complete graphs using local rewiring operations only. The probability distribution of graphs of this kind is shown to depend on the…

We propose a slightly revised Miller-Hagberg (MH) algorithm that efficiently generates a random network from a given expected degree sequence. The revision was to replace the approximated edge probability between a pair of nodes with a…

Seymour's second neighbourhood conjecture asserts that every oriented graph has a vertex whose second out-neighbourhood is at least as large as its out-neighbourhood. In this paper, we prove that the conjecture holds for quasi-transitive…

We introduce the Parikh-de-Bruijn grid, a graph whose vertices are fixed-order Parikh vectors, and whose edges are given by a simple shift operation. This graph gives structural insight into the nature of sets of Parikh vectors as well as…

Isoperimetric inequalities form a very intuitive yet powerful characterization of the connectedness of a state space, that has proven successful in obtaining convergence bounds. Since the seventies they form an essential tool in…

We are interested in the relationships between the number fixed points in a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ and its interaction graph, which is the arc-signed digraph $G$ on $\{1,\dots,n\}$ that describes the positive and negative…

Let $k\geq3$ be a fixed integer and let $Z_k(G)$ be the number of $k$-colourings of the graph $G$. For certain values of the average degree, the random variable $Z_k(G(n,m))$ is known to be concentrated in the sense that $\frac1n(\ln…