Related papers: Secure Frameproof Code Through Biclique Cover

The lettericity of a graph $G=(V,E)$ is defined as the smallest size of an alphabet $\Sigma$ such that there is a word $w_1 \dots w_{|V|} \in \Sigma^*$ and a decoder $\mathcal{D} \subseteq \Sigma^2$ with the property that $G$ is isomorphic…

A conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is $3$-edge-colourable, the rest of cubic…

It is shown that the maximum size of a binary subspace code of packet length $v=6$, minimum subspace distance $d=4$, and constant dimension $k=3$ is $M=77$; in Finite Geometry terms, the maximum number of planes in $\operatorname{PG}(5,2)$…

We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code;…

Frameproof codes are a class of secure codes that were originally introduced in the pioneering work of Boneh and Shaw in the context of digital fingerprinting. They can be used to enhance the security and credibility of digital content. Let…

For any positive integer $r$, an $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and pairwise distinct. For a…

A signed graph is a graph $G$ associated with a mapping $\sigma: E(G)\to \{-1,+1\}$, denoted by $(G,\sigma)$. A $cycle$ of $(G,\sigma)$ is a connected 2-regular subgraph. A cycle $C$ is $positive$ if it has an even number of negative edges,…

If $S$ is a transitive metric space, then $|C|\cdot|A| \le |S|$ for any distance-$d$ code $C$ and a set $A$, ``anticode'', of diameter less than $d$. For every Steiner S$(t,k,n)$ system $S$, we show the existence of a $q$-ary…

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ that is an independent set such that every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A total perfect code in $\Gamma$ is a subset $C$ of $V$ such…

There are several topological results ensuring the existence of a large complete bipartite subgraph in any properly colored graph satisfying some special topological regularity conditions. In view of $\mathbb{Z}_p$-Tucker lemma, Alishahi…

A 1978 theorem of Kozen states that two graphs on $n$ vertices are isomorphic if and only if there is a clique of size $n$ in the weak modular product between the two graphs. Restricting to bipartite graphs and considering complete…

Our main result is that every graph $G$ on $n\ge 10^4r^3$ vertices with minimum degree $\delta(G) \ge (1 - 1 / 10^4 r^{3/2} ) n$ has a fractional $K_r$-decomposition. Combining this result with recent work of Barber, K\"uhn, Lo and Osthus…

Given a family of 3-graphs F its codegree threshold coex(n, F) is the largest number d=d(n) such that there exists an n-vertex 3-graph in which every pair of vertices is contained in at least d 3-edges but which contains no member of F as a…

The twin-width of a graph $G$ is the minimum integer $d$ such that $G$ has a $d$-contraction sequence, that is, a sequence of $|V(G)|-1$ iterated vertex identifications for which the overall maximum number of red edges incident to a single…

We estimate the maximum possible number of cliques of size $r$ in an $n$-vertex graph free of a fixed complete $r$-partite graph $K_{s_1, s_2, \ldots, s_r}$. By viewing every $r$-clique as a hyperedge, the upper bound on the Tur\'an number…

For a graph $G$, a hypergraph $\mathcal{H}$ is a Berge copy of $G$ (or a Berge-$G$ in short), if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for each $e \in E(G)$ we have $e \subseteq f(e)$. We denote the family of…

In the vertex cover problem, the input is a graph $G$ and an integer $k$, and the goal is to decide whether there is a set of vertices $S$ of size at most $k$ such that every edge of $G$ is incident on at least one vertex in $S$. We study…

A graph $G$ is {\em well-covered} if every maximal independent set has the same cardinality $q$. Let $i_k(G)$ denote the number of independent sets of cardinality $k$ in $G$. Brown, Dilcher, and Nowakowski conjectured that the independence…

Kostochka and Yancey resolved a famous conjecture of Ore on the asymptotic density of $k$-critical graphs by proving that every $k$-critical graph $G$ satisfies $|E(G)| \geq (\frac{k}{2} - \frac{1}{k-1})|V(G)| - \frac{k(k-3)}{2(k-1)}$. The…

We introduce a new class of rate one-half binary codes: {\bf complementary information set codes.} A binary linear code of length $2n$ and dimension $n$ is called a complementary information set code (CIS code for short) if it has two…