Related papers: Solving a Conjecture on Identification in Hamming …

It is known that the minimum broadcast rate of a linear index code over $\mathbb{F}_q$ is equal to the $minrank_q$ of the underlying digraph. In [3] it is proved that for $\mathbb{F}_2$ and any positive integer $k$, $minrank_q(G)\leq k$ iff…

Strengthening Hadwiger's conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$-minor is properly $(t-1)$-colorable, this is known as the Odd Hadwiger's conjecture. We prove a relaxation of the above…

For smooth mappings of the unit disc into the oriented Grassmannian manifold $\mathbb G_{n,2}$, H\'elein (2002) conjectured the global existence of Coulomb frames with bounded conformal factor provided the integral of $|\boldsymbol A|^2$,…

Given a hypergraph $H$ and a weight function $w: V \rightarrow \{1, \dots, M\}$ on its vertices, we say that $w$ is isolating if there is exactly one edge of minimum weight $w(e) = \sum_{i \in e} w(i)$. The Isolation Lemma is a…

We study the problem of counting $k$-hypergraphlets, an interesting but surprisingly ignored primitive, with the aim of understanding whether efficient algorithms exist. To this end, we consider color coding, a well-known technique for…

The problems of detecting and recovering planted structures/subgraphs in Erd\H{o}s-R\'{e}nyi random graphs, have received significant attention over the past three decades, leading to many exciting results and mathematical techniques.…

Let $G$ be a graph of order $n$. A classical upper bound for the domination number of a graph $G$ having no isolated vertices is $\lfloor\frac{n}{2}\rfloor$. However, for several families of graphs, we have $\gamma(G) \le…

An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We show a dichotomy for the size of the smallest identifying code in classes…

Let $F^n$ be the binary $n$-cube, or binary Hamming space of dimension $n$, endowed with the Hamming distance, and ${\cal E}^n$ (respectively, ${\cal O}^n$) the set of vectors with even (respectively, odd) weight. For $r\geq 1$ and $x\in…

Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a…

In this paper, we study a special kind of factorization of $x^n+1$ over $\mathbb{F}_q, $ with $q$ a prime power $\equiv 3~({\rm mod}~4)$ when $n=2p,$ with $p\equiv 3~({\rm mod}~4)$ and $p$ is a prime. Given such a $q$ infinitely many such…

Since the seminal result of Karger, Motwani, and Sudan, algorithms for approximate 3-coloring have primarily centered around SDP-based rounding. However, it is likely that important combinatorial or algebraic insights are needed in order to…

In 1995, Brass, Harborth and Nienborg disproved a conjecture of Erd\H{o}s when they showed that a $C_4$-free subgraph of the hypercube, $Q_n$, can have at least $(\frac 12 +\omega(1))e(Q_n)$ edges. In this paper, we generalize the idea of…

For an integer $q\ge 2$, a perfect $q$-hash code $C$ is a block code over $[q]:=\{1,\ldots,q\}$ of length $n$ in which every subset $\{\mathbf{c}_1,\mathbf{c}_2,\dots,\mathbf{c}_q\}$ of $q$ elements is separated, i.e., there exists…

We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing…

Let $\text{ch}(G)$ denote the choice number of a graph $G$ (also called "list chromatic number" or "choosability" of $G$). Noel, Reed, and Wu proved the conjecture of Ohba that $\text{ch}(G)=\chi(G)$ when $|V(G)|\le 2\chi(G)+1$. We extend…

We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming…

The graph homomorphism problem (HOM) asks whether the vertices of a given $n$-vertex graph $G$ can be mapped to the vertices of a given $h$-vertex graph $H$ such that each edge of $G$ is mapped to an edge of $H$. The problem generalizes the…

For a positive integer $k$ and a graph $H$ on $k$ vertices, we are interested in the inducibility of $H$, denoted $\mathrm{ind}(H)$, which is defined as the maximum possible probability that choosing $k$ vertices uniformly at random from a…

We study the problem of recognizing graph powers and computing roots of graphs. We provide a polynomial time recognition algorithm for r-th powers of graphs of girth at least 2r+3, thus improving a bound conjectured by Farzad et al. (STACS…