Related papers: Minimum de Bruijn Sequence in a Language with Forb…

We give two combinatorial proofs of the fact that the number of loopless digraphs on the vertex set $[n]$ with no isolated vertices and with exactly one Eulerian tour up to a cyclic shift is $\frac{1}{2}(n-1)!C_{n}$, where $C_{n}$ denotes…

We propose an efficient linear-time graph-based divisive cluster analysis approach called Reductive Clustering. The approach tries to reveal the hierarchical structural information through reducing the graph into a more concise one…

Decoder-only transformers lead to a step-change in capability of large language models. However, opinions are mixed as to whether they are really planning or reasoning. A path to making progress in this direction is to study the model's…

In a \emph{rotor walk} the exits from each vertex follow a prescribed periodic sequence. On an infinite Eulerian graph embedded periodically in $\R^d$, we show that any simple rotor walk, regardless of rotor mechanism or initial rotor…

A longest path in a graph is called a detour. It is easy to see that a connected graph of minimum degree at least $2$ and order at least $4$ has at least $4$ detours. We prove that if the number of detours in such a graph of order at least…

The spectra of digraphs, unlike those of graphs, is a relatively unexplored territory. In a digraph, a separation is a pair of sets of vertices X and Y such that there are no arcs from X and Y . For a subclass of eulerian digraphs, we give…

We revisit results obtained in [F. Harary, U. Peled, Hamiltonian threshold graphs, Discrete Appl.~Math., 16 (1987), 11--15], where several necessary and necessary and sufficient conditions for a connected threshold graph to be Hamiltonian…

We show that every directed graph with minimum out-degree at least $18k$ contains at least $k$ vertex disjoint cycles. This is an improvement over the result of Alon who showed this result for digraphs of minimum out-degree at least $64k$.…

Let $k_r(n,\delta)$ be the minimum number of $r$-cliques in graphs with $n$ vertices and minimum degree $\delta$. We evaluate $k_r(n,\delta)$ for $\delta \leq 4n/5$ and some other cases. Moreover, we give a construction, which we conjecture…

We study separating systems of the edges of a graph where each member of the separating system is a path. We conjecture that every $n$-vertex graph admits a separating path system of size $O(n)$ and prove this in certain interesting special…

A brick is a 3-connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. A brick is minimal if for every edge e the deletion of e results in a graph that is not a brick. We prove a…

The de Bruijn torus (or grid) problem looks to find an $n$-by-$m$ binary matrix in which every possible $j$-by-$k$ submatrix appears exactly once. The existence and construction of these binary matrices was determined in the 70's, with…

We apply model theoretic methods to the problem of existence of countable universal graphs with finitely many forbidden connected subgraphs. We show that to a large extent the question reduces to one of local finiteness of an…

Let $\widetilde{\alpha}$ be a length-$L$ cyclic sequence of characters from a size-$K$ alphabet $\mathcal{A}$ such that the number of occurrences of any length-$m$ string on $\mathcal{A}$ as a substring of $\widetilde{\alpha}$ is $\lfloor L…

For an arbitrary finite family of graphs, the distance labeling problem asks to assign labels to all nodes of every graph in the family in a way that allows one to recover the distance between any two nodes of any graph from their labels.…

Minimal separators in graphs are an important concept in algorithmic graph theory. In particular, many problems that are NP-hard for general graphs are known to become polynomial-time solvable for classes of graphs with a polynomially…

We consider the problem of inferring an edge-labeled graph from the sequence of edge labels seen in a walk of that graph. It has been known that this problem is solvable in $O(n \log n)$ time when the targets are path or cycle graphs. This…

In this paper we introduce a new parameter for a graph called the {\it minimum universal rank}. This parameter is similar to the minimum rank of a graph. For a graph $G$ the minimum universal rank of $G$ is the minimum rank over all…

A graph $G$ is called a sum graph if there is a so-called sum labeling of $G$, i.e. an injective function $\ell: V(G) \rightarrow \mathbb{N}$ such that for every $u,v\in V(G)$ it holds that $uv\in E(G)$ if and only if there exists a vertex…

In trying to generalize the classic Sylvester-Gallai theorem and De Bruijn-Erd\H{o}s theorem in plane geometry, lines and closure lines were previously defined for metric spaces and hypergraphs. Both definitions do not obey the geometric…