Related papers: Une conjecture de Kontsevich et Shoikhet et la car…
Turaev's shadow formula calculates the SU(2)-Reshetikhin-Turaev-Witten invariants using shadows, and its expression is somehow similar to a Euler characteristic. We give a short proof of this formula using skein theory. The formula applies…
A beautiful conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same…
This is a companion note to our paper 'Some advances on Sidorenko's conjecture', elaborating on a remark in that paper that the approach which proves Sidorenko's conjecture for strongly tree-decomposable graphs may be extended to a broader…
We show that Lusztig's conjecture on the irreducible characters of a reductive algebraic group over a field of positive characteristic is equivalent to the generic multiplicity conjecture, which gives a formula for the Jordan-H"older…
We propose a Law of Nature? Viz., Pure Regularity Occurs at Na\"ive Levels and Regularity has Affinity with Evenness. In a series of three papers, it was established that regular Euler graphs with only one type of (pure) cycles are…
In this paper, we are motivated by the conjectures proposed by C.~Bender \textit{et al.}, \cite{C} in 2024. We have settled the first two conjectures negatively by providing a counter example in \cite{KTJ}, whereas in this paper, we prove…
The Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was…
In 2019, P. Higgins formulated [1] a question about bipartite graphs (see Conjecture 1 below); this question arises in the study of regular finite semigroups. F. V. Petrov formulated [2] another combinatorial conjecture (Conjecture 3);…
We define Strebel differentials for stable complex curves, prove the existence and uniqueness theorem that generalizes Strebel's theorem for smooth curves, prove that Strebel differentials form a continuous family over the moduli space of…
For a graph $G$, the $k$-colouring graph of $G$ has vertices corresponding to proper $k$-colourings of $G$ and edges between colourings that differ at a single vertex. The graph supports the Glauber dynamics Markov chain for $k$-colourings,…
We exhibit a 5-uniform hypergraph that has no polychromatic 3-coloring, but all its restricted subhypergraphs with edges of size at least 3 are 2-colorable. This disproves a bold conjecture of Keszegh and the author, and can be considered…
In this article, we give an elementary combinatorial proof of a conjecture about the determination of automorphism group of the power graph of finite cyclic groups, proposed by Doostabadi, Erfanian and Jafarzadeh in 2013.
Let $G$ be a graph with adjacency matrix $A(G)$. We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where $n^+(G)$ and $n^-(G)$ denote the number of positive and negative eigenvalues of $A(G)$, respectively. This conjecture generalizes…
In the paper [J. Graph Theory (2023) 102:458-471, the Esperet's conjecture has been posed: Every $\chi$-bounded hereditary class is poly-$\chi$-bounded]. This conjecture was first posed in [Habilitation Thesis, Universit\'e Grenoble Alpes,…
By finding orthogonal representation for a family of simple connected called $\delta$-graphs it is possible to show that $\delta$-graphs satisfy delta conjecture. An extension of the argument to graphs of the form…
It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field k and any finite set of places S of k, one can effectively compute the set of isomorphism classes of…
It was independently conjectured by H\"aggkvist in 1989 and Kriesell in 2011 that given a positive integer $\ell$, every simple eulerian graph with high minimum degree (depending on $\ell$) admits an eulerian tour such that every segment of…
The Tate conjecture has two parts: an assertion (S) about semisimplicity of Galois representations, and an assertion (T) which says that every Tate class is algebraic. We show that in characteristic 0, (T) implies (S). In characteristic p…
To every tree we associate a filtered cochain complex. Its cohomology and the corresponding spectral sequence have clear combinatorial description. If a tree is the Dynkin diagram of a simple plane curve singularity, the graded Euler…
A 3-graph $\mathcal{F}$ is \emph{$U(s, 2s+1)$} if for any $s$ edges $e_1,...,e_s\in E(\mathcal{F})$, $|e_1\cup...\cup e_s|\leq 2s+1$. Frankl and Kupavskii (2020) proposed the following conjecture: For any $3$-graph $\mathcal{F}$ with $n$…