Related papers: Cycles of given size in a dense graph

A $3$-uniform loose cycle, denoted by $C_t$, is a $3$-graph on $t$ vertices whose vertices can be arranged cyclically so that each hyperedge consists of three consecutive vertices, and any two consecutive hyperedges share exactly one…

Consider a directed multigraph $D$ that is balanced (i.e., at each vertex, the indegree equals the outdegree). Let $A$ be its set of arcs. Fix an integer $k$. Let $s$ be a vertex of $D$. We show that the number of $k$-element subsets $B$ of…

We devise constant-factor approximation algorithms for finding as many disjoint cycles as possible from a certain family of cycles in a given planar or bounded-genus graph. Here disjoint can mean vertex-disjoint or edge-disjoint, and the…

Let $n\geq k\geq r+3$ and $\mathcal H$ be an $n$-vertex $r$-uniform hypergraph. We show that if $|\mathcal H|> \frac{n-1}{k-2}\binom{k-1}{r}$ then $\mathcal H$ contains a Berge cycle of length at least $k$. This bound is tight when $k-2$…

We address a problem posed by Erd\H{o}s and Hajnal in 1991, proving that for all $n \geq 600$, every $(2n+1)$-vertex graph with at least $n^2 + n + 1$ edges contains two vertices of equal degree connected by a path of length three. The…

In this paper we provide some sufficient conditions for the existence of an odd or even cycle that passing a given vertex or an edge in $2$-connected or $2$-edge connected graphs. We provide some similar conditions for the existence of an…

Let $G$ be a graph and $k \geq 3$ an integer. A subset $D \subseteq V(G)$ is a $k$-clique (resp., cycle) isolating set of $G$ if $G-N[D]$ contains no $k$-clique (resp., cycle). In this paper, we prove that every connected graph with maximum…

Every 4-connected graph with minimum degree $\delta$ and connectivity $\kappa$ either has a cycle of length at least $4\delta-2\kappa$ or has a dominating cycle.

We extend the study of link-irregular graphs to directed graphs (digraphs), where a digraph is link-irregular if no two vertices have isomorphic directed links. We establish that link-irregular digraphs exist on $n$ vertices if and only if…

The analysis of large simple graphs with extreme values of the densities of edges and triangles has been extended to the statistical structure of typical graphs of fixed intermediate densities, by the use of large deviations of Erdoes-Renyi…

Let $G$ be a graph, and $v\in V(G)$ and $S\subseteq V(G)\backslash v$ of size at least $k$. An important result on graph connectivity due to Perfect states that, if $v$ and $S$ are $k$-linked, then a $(k-1)$-link between a vertex $v$ and…

In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every \alpha > 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8…

The Erd\H{o}s--Gallai Theorem states that for $k \geq 3$, any $n$-vertex graph with no cycle of length at least $k$ has at most $\frac{1}{2}(k-1)(n-1)$ edges. A stronger version of the Erd\H{o}s--Gallai Theorem was given by Kopylov: If $G$…

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…

Dirac (1952) proved that every connected graph of order $n>2k+1$ with minimum degree more than $k$ contains a path of length at least $2k+1$. Erd\H{o}s and Gallai (1959) showed that every $n$-vertex graph $G$ with average degree more than…

Consider a family of graphs having a fixed girth and a large size. We give an optimal lower asymptotic bound on the number of even cycles of any constant length, as the order of the graphs tends to infinity.

For a positive integer $k\ge 1$, a graph $G$ is $k$-stepwise irregular ($k$-SI graph) if the degrees of every pair of adjacent vertices differ by exactly $k$. Such graphs are necessarily bipartite. Using graph products it is demonstrated…

There has been extensive research on cycle lengths in graphs with large minimum degree. In this paper, we obtain several new and tight results in this area. Let $G$ be a graph with minimum degree at least $k+1$. We prove that if $G$ is…

We prove that for any given $\varepsilon>0$ and $d\in [0,1]$, every sufficiently large $(\varepsilon, d)$-dense graph $G$ contains for each odd integer $r$ at least $(d^r-\varepsilon)|V(G)|^r$ cycles of length $r$. Here, $G$ being…

A famous conjecture of P\'osa from 1962 asserts that every graph on $n$ vertices and with minimum degree at least $2n/3$ contains the square of a Hamilton cycle. The conjecture was proven for large graphs in 1996 by Koml\'os, S\'ark\"ozy…