Related papers: The shape of $x^2\bmod n$
In this paper, we explore some generalizations of a counting problem related to tilings in grids of size 2xn, which was originally posed as a question on Mathematics Stack Exchange (Question 3972905). In particular, we consider this problem…
In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces, and maps of graphs embedded in the sphere, in homotopy type theory. This allows us to provide an elementary…
The Tur\'an number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. Let P_k be the path with k vertices, the square P^2_k of P_k is obtained by joining the pairs of vertices…
We survey the construction of the Cox ring of an algebraic variety X and study the birational geometry of X when its Cox ring is finitely generated.
The Mapper algorithm is a popular tool for visualization and data exploration in topological data analysis. We investigate an inverse problem for the Mapper algorithm: Given a dataset $X$ and a graph $G$, does there exist a set of Mapper…
We introduce a new notation for representing labeled regular bipartite graphs of arbitrary degree. Several enumeration problems for labeled and unlabeled regular bipartite graphs have been introduced. A general algorithm for enumerating all…
We examine the family of nestohedra resulting from the complete bipartite graph through the medium of a generating function and demonstrate some of their combinatorial invariants.
We enumerate the independent sets of several classes of regular and almost regular graphs and compute the corresponding generating functions. We also note the relations between these graphs and other combinatorial objects and, in some…
We discuss various aspects of the statistical formulation of the theory of random graphs, with emphasis on results obtained in a series of our recent publications.
We study homeomorphisms of controlled $p$-module by certain integrals. In this way, we establish various properties of mappings and show that their features are close to quasiconformal and bilipschitz mappings.
We give a sharp bound for the automorphism group of a cubic simple graph with a given number of vertices. For each number of vertices we give an explicit graph attaining the bound, and prove its uniqueness in special cases.
Firstly, for a general graph, we find a recursion formula on the number of Hamiltonian cycles and one on cycles. By this result, we give some new polynomial invariants. Secondly, we give a condition to tell whether a polynomial defined by…
We study the convergence of graphs consisting of finitely many internal rays for degenerating Newton maps. We state a sufficient condition to guarantee the convergence. As an application, we investigate the boundedness of hyperbolic…
Topics concerning metric dimension related invariants in graphs are nowadays intensively studied. This compendium of combinatorial and computational results on this topic is an attempt of surveying those contributions that are of the…
We prove almost tight bounds on the length of paths in $2$-edge-connected cubic graphs. Concretely, we show that (i) every $2$-edge-connected cubic graph of size $n$ has a path of length $\Omega\left(\frac{\log^2{n}}{\log{\log{n}}}\right)$,…
This study extends a prior investigation of limit shapes for partitions of integers, which was based on analysis of sums of geometric random variables. Here we compute limit shapes for grand canonical Gibbs ensembles of partitions of sets,…
A new kind of diagrams is presented, showing the causal structure of bimetric interactions.
In this paper we present a novel framework for sequence to shape maps. These combinatorial maps realize exponentially many shapes, and have preimages which contain extended connected subgraphs of diameter n (neutral networks). We prove that…
Let $G$ be a connected undirected graph on $n$ vertices with no loops but possibly multiedges. Given an arithmetical structure $(\textbf{r}, \textbf{d})$ on $G$, we describe a construction which associates to it a graph $G'$ on $n-1$…
The famous Gallai's Conjecture states that any connected graph with n vertices has a path decomposition containing at most (n+1)/2 paths. In this note, we explore graphs generated from removing edges from complete graphs. We first provide…