Related papers: Minimizing cycles in tournaments and normalized $q…

In this paper, we prove a tight minimum degree condition in general graphs for the existence of paths between two given endpoints, whose lengths form a long arithmetic progression with common difference one or two. This allows us to obtain…

There is a rich history of studying the existence of cycles in planar graphs. The famous Tutte theorem on the Hamilton cycle states that every 4-connected planar graph contains a Hamilton cycle. Later on, Thomassen (1983), Thomas and Yu…

Reid conjectured that any finite set of non-negative integers is the score set of some tournament and Yao gave a non-constructive proof of Reid's conjecture using arithmetic arguments. No constructive proof has been found since. In this…

In 2022, Gao, Huo, Liu, and Ma proved that every graph with minimum degree at least $k+1$ contains $k$ admissible cycles, where a set of $k$ cycles is said to be admissible if their lengths form an arithmetic progression with common…

The investigation of conditions guaranteeing the appearance of cycles of certain lengths is one of the most well-studied topics in graph theory. In this paper we consider a problem of this type which asks, for fixed integers ${\ell}$ and…

Mantel's theorem states that every $n$-vertex graph with $\lfloor \frac{n^2}{4} \rfloor +t$ edges, where $t>0$, contains a triangle. The problem of determining the minimum number of triangles in such a graph is usually referred to as the…

For $1\le d\le \ell< k$, we give a new lower bound for the minimum $d$-degree threshold that guarantees a Hamilton $\ell$-cycle in $k$-uniform hypergraphs. When $k\ge 4$ and $d< \ell=k-1$, this bound is larger than the conjectured minimum…

We study the number and distribution of the limit cycles of a planar vector field whose component functions are random polynomials. We prove a lower bound on the average number of limit cycles when the random polynomials are sampled from…

The Erd\H{o}s-Hajnal Conjecture states that for every $H$ there exists a constant $\epsilon(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least…

Given $k\ge3$ and $1\leq \ell< k$, an $(\ell,k)$-cycle is one in which consecutive edges, each of size $k$, overlap in exactly $\ell$ vertices. We study the smallest number of edges in $k$-uniform $n$-vertex hypergraphs which do not contain…

For two integers $k$ and $\ell$, an $(\ell \text{ mod }k)$-cycle means a cycle of length $m$ such that $m\equiv \ell\pmod{k}$. In 1977, Bollob\'{a}s proved a conjecture of Burr and Erd\H{o}s by showing that if $\ell$ is even or $k$ is odd,…

In this paper, we study the distribution of the number of occurrences of the simplest frame pattern, called the $\mu$ pattern, in $n$-cycles. Given an $n$-cycle $C$, we say that a pair $\langle i,j \rangle$ matches the $\mu$ pattern if $i <…

Burr and Erd\H{o}s conjectured that for each $k,\ell \in \mathbb Z^+$ such that $k \mathbb Z + \ell$ contains even integers, there exists $c_k(\ell)$ such that any graph of average degree at least $c_k(\ell)$ contains a cycle of length…

Nearly-doubly-regular tournaments have played significant roles in extremal graph theory. In this note, we construct new cyclotomic nearly-doubly-regular tournaments and determine their spectrum by establishing a new connection between…

A distinguishing $r$-labeling of a digraph $G$ is a mapping $\lambda$ from the set of verticesof $G$ to the set of labels $\{1,\dots,r\}$ such that no nontrivial automorphism of $G$ preserves all the labels.The distinguishing number $D(G)$…

We prove moderate deviations bounds for the lower tail of the number of odd cycles in a $\calG(n, m)$ random graph. We show that the probability of decreasing triangle density by $t^3$, is $\exp(-\Theta(n^2 t^2))$ whenever $n^{-3/4} \ll t^3…

Burr and Erd\H{o}s conjectured in 1976 that for every two integers $k>\ell\geqslant 0$ satisfying that $k\mathbb{Z}+\ell$ contains an even integer, an $n$-vertex graph containing no cycles of length $\ell$ modulo $k$ can contain at most a…

Liu and Ma [J. Combin. Theory Ser. B, 2018] conjectured that every $2$-connected non-bipartite graph with minimum degree at least $k+1$ contains $\lceil k/2\rceil $ cycles with consecutive odd lengths. In particular, they showed that this…

For each $k \geq 3$ and $1 \leq \ell \leq k-1$ we give an asymptotically best possible minimum positive codegree condition for the existence of a Hamilton $\ell$-cycle in a $k$-uniform hypergraph. This result exhibits an interesting duality…

We consider the manipulability of tournament rules which map the results of $\binom{n}{2}$ pairwise matches and select a winner. Prior work designs simple tournament rules such that no pair of teams can manipulate the outcome of their match…