Related papers: Van Lint-MacWilliams' conjecture and maximum cliqu…

We determine new upper bounds for the clique numbers of strongly regular graphs in terms of their parameters. These bounds improve on the Delsarte bound for infinitely many feasible parameter tuples for strongly regular graphs, including…

Let \mathcal{F}_k denote the family of 2-edge-colored complete graphs on 2k vertices in which one color forms either a clique of order k or two disjoint cliques of order k. Bollob\'as conjectured that for every \epsilon>0 and positive…

We study the fine-grained complexity of evaluating Boolean Conjunctive Queries and their generalization to sum-of-product problems over an arbitrary semiring. For these problems, we present a general semiring-oblivious reduction from the…

A well-known conjecture of Erd\H{o}s and S\'os states that every graph with average degree exceeding $m-1$ contains every tree with $m$ edges as a subgraph. We propose a variant of this conjecture, which states that every graph of maximum…

We construct classes of graphs that are variants of the so-called layered wheel. One of their key properties is that while the treewidth is bounded by a function of the clique number, the construction can be adjusted to make the dependance…

In this note we prove that all connected Cayley graphs of every finite group $Q \times H$ are 1-factorizable, where $Q$ is any non-trivial group of 2-power order and $H$ is any group of odd order.

The celebrated Erd\H{o}s-Ko-Rado (EKR) theorem for Paley graphs (of square order) states that all maximum cliques are canonical in the sense that each maximum clique arises from the subfield construction. Recently, Asgarli and Yip extended…

The generating graph $\Gamma(G)$ of a finite group $G$ has vertex set the non-identity elements of $G$, with two elements connected exactly when they generate $G$. A coclique in a graph is an empty induced subgraph, so a coclique in…

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for \textsc{Maximum Clique} on unit disk graphs [Clark, Colbourn, Johnson; Discrete…

The Brouwer conjecture (BC) in spectral graph theory claims that the sum of the largest k Kirchhoff eigenvalues of a graph are bounded above by the number m of edges plus k(k+1)/2. We show that (BC) holds for all graphs with n vertices if n…

Suppose that $E$ is an elliptic curve defined over $\mathbb{Q}$ without complex multiplication and with conductor $N$. For each positive integer $m$, the action of the absolute Galois group…

We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang's Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures…

We show that there is an absolute constant $c>0$ such that every large connected $n$-vertex Cayley graph with degree $d\geq n^{1-c}$ has a Hamilton cycle. This makes progress towards the Lov\'asz conjecture and improves upon the previous…

In this paper we consider $r$-regular graphs $G$ that admit the vertex set partition such that one of the induced subgraphs is the join of an $s$-vertex clique and a $t$-vertex co-clique and represents a star complement for an eigenvalue…

Bondy and Vince proved that a graph of minimum degree at least three contains two cycles whose lengths differ by one or two, which was conjectured by Erd\H{o}s. Gao, Li, Ma and Xie gave an average degree counterpart of Bondy-Vince's result,…

For integers $r,t\geq2$ and $n\geq1$ let $f_r(t,n)$ be the minimum, over all factorizations of the complete $r$-uniform hypergraph of order $n$ into $t$ factors $H_1,\dots,H_t$, of $\sum_{i=1}^tc(H_i)$ where $c(H_i)$ is the number of…

For a prime power $q$ and a positive integer $n$, we say a subspace $U$ of ${\mathbb{F}_q^n}$ is {\em cyclically covering} if the union of the cyclic shifts of $U$ is equal to $\mathbb{F}_q^n$. We investigate the problem of determining the…

A simple graph $G$ is \emph{overfull} if $|E(G)|>\Delta\lfloor|V(G)|/2\rfloor$. By the pigeonhole principle, every overfull graph $G$ has $\chi'(G)>\Delta$. The \emph{core} of a graph, denoted $G_\Delta$, is the subgraph induced by its…

We show that the cop number of the Cayley sum graph of a finite group $G$ with respect to a symmetric subset $S$ is at most twice its degree when the graph is connected, undirected. We also prove that a similar bound holds for the cop…

The P\'osa-Seymour conjecture asserts that every graph on $n$ vertices with minimum degree at least $(1 - 1/(r+1))n$ contains the $r^{th}$ power of a Hamilton cycle. Koml\'os, S\'ark\"ozy and Szemer\'edi famously proved the conjecture for…