Related papers: Cycles on a multiset with only even-odd drops

In this paper we study cycles in random bipartite graph $G(n,n,p)$. We prove that if $p\gg n^{-2/3}$, then $G(n,n,p)$ a.a.s. satisfies the following. Every subgraph $G'\subset G(n,n,p)$ with more than $(1+o(1))n^2p/2$ edges contains a cycle…

We show that provided $\log^{50} n/n \leq p \leq 1 - n^{-1/4}\log^9 n$ we can with high probability find a collection of $\lfloor \delta(G)/2 \rfloor$ edge-disjoint Hamilton cycles in $G \sim G_{n, p}$, plus an additional edge-disjoint…

A permutation is defined to be cycle-up-down if it is a product of cycles that, when written starting with their smallest element, have an up-down pattern. We prove bijectively and analytically that these permutations are enumerated by the…

Let $F$ be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of $F$ in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits,…

Using a result of Gessel and Reutenauer, we find a simple formula for the number of cyclic permutations with a given descent set, by expressing it in terms of ordinary descent numbers (i.e., those counting all permutations with a given…

Li, Nikiforov and Schelp conjectured that a 2-edge coloured graph G with order n and minimal degree strictly greater than 3n/4 contains a monochromatic cycle of length l, for all l at least four and at most n/2. We prove this conjecture for…

A conjecture of Birmel\'e, Bondy and Reed states that for any integer $\ell\geq 3$, every graph $G$ without two vertex-disjoint cycles of length at least $\ell$ contains a set of at most $\ell$ vertices which meets all cycles of length at…

The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller…

We define a tournament to be alternation acyclic if it does not contain a cycle in which descents and ascents alternate. Using a result by Athanasiadis on hyperplane arrangements, we show that these tournaments are counted by the median…

Consider the collection of all t-multisets of {1,...,n}. A universal cycle on multisets is a string of numbers, each of which is between 1 and n, such that if these numbers are considered in t-sized windows, every multiset in the collection…

Let $G$ be a 2-connected $n$-vertex graph and $N_s(G)$ be the total number of $s$-cliques in $G$. Let $k\ge 4$ and $s\ge 2$ be integers. In this paper, we show that if $G$ has an edge $e$ which is not on any cycle of length at least $k$,…

A Universal Cycle for t-multisets of [n]={1,...,n} is a cyclic sequence of $\binom{n+t-1}{t}$ integers from [n] with the property that each t-multiset of [n] appears exactly once consecutively in the sequence. For such a sequence to exist…

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…

We analyze the dynamics of a class of $\mathbb{Z}_{2n}$-equivariant differential equations on the plane, depending on 4 real parameters. This study is the generalisation to $\mathbb{Z}_{2n}$ of previous works with $\mathbb{Z}_4$ and…

Let $H$ be a fixed graph and $\mathcal{G}$ a subcritical graph class. In this paper we show that the number of occurrences of $H$ (as a subgraph) in a uniformly at random graph of size $n$ in $\mathcal{G}$ follows a normal limiting…

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…

The components of the graphs $D(n, q)$ provide the best-known general lower bound for the number of edges in a graph with $n$ vertices and no cycles of length less than $g$. In this paper, we give a new, short, and simpler proof of the fact…

In 1962 P\'osa conjectured that every graph G on n vertices with minimum degree at least 2n/3 contains the square of a hamiltonian cycle. In 1996 Fan and Kierstead proved the path version of P\'osa's Conjecture. They also proved that it…

In this short note it is shown that every graph of diameter 2 and minimum degree at least 3 contains a cycle of length 4 or 8. This result contributes to the study of the Erd\H{o}s-Gy\'arf\'as Conjecture by confirming it for the class of…

In this paper, we introduce a method of constructing Universal Cycles on sets by taking "sums" and "products" of smaller cycles. We demonstrate this new approach by proving that if there exist Universal Cycles on the 4-subsets of [18] and…