Related papers: Cycle-factors in oriented graphs

A recent result by Kardo\v{s}, M\'a\v{c}ajov\'a and Zerafa [J. Comb. Theory, Ser. B. 160 (2023) 1--14] related to the famous Berge-Fulkerson conjecture implies that given an arbitrary set of odd pairwise edge-disjoint cycles, say $\mathcal…

Gallai asked in 1984 if any $k$-critical graph on $n$ vertices contains at least $n$ distinct $(k-1)$-critical subgraphs. The answer is trivial for $k\leq 3$. Improving a result of Stiebitz, Abbott and Zhou proved in 1995 that for all…

Christoph, Dragani\'{c}, Gir\~{a}o, Hurley, Michel, and M\"{u}yesser conjectured that, when $d\mid n$, the expected number of cycles in a uniformly random cycle-factor of a directed $d$-regular graph on $n$ vertices is uniquely maximised by…

We prove that for all natural numbers $m$ and $k$ where $k$ is odd, there exists a natural number $N(k)$ such that any 3-connected cubic graph with at least $N(k)$ vertices contains a cycle of length $m$ modulo $k$. We also construct a…

We conjecture that a 2-connected graph $G$ of order $n$, in which $d(x)+d(y)\geq n-k$ for every pair of non-adjacent vertices $x$ and $y$, contains a cycle of length $n-k$ ($k<n/2$), unless $G$ is bipartite and $n-k$ is odd. This…

A cycle C(k1,k2,...,kn) is the oriented cycle formed of n blocks of lengths k1,k2,...,kn-1 and kn respectively. In 2018 Cohen et al. conjectured that for every positive integers k1,k2,...,kn there exists a constant g(k1,k2,...,kn) such that…

A new efficient algorithm is presented for finding all simple cycles that satisfy a length constraint in a directed graph. When the number of vertices is non-trivial, most cycle-finding problems are of practical interest for sparse graphs…

We prove several results concerning cycle tilings and $H$-factors in digraphs. We provide a minimum semi-degree condition for forcing a digraph to contain a given spanning collection of vertex-disjoint orientations of cycles. Our result is…

We show that for each \ell\geq 4 every sufficiently large oriented graph G with \delta^+(G), \delta^-(G) \geq \lfloor |G|/3 \rfloor +1 contains an \ell-cycle. This is best possible for all those \ell\geq 4 which are not divisible by 3.…

Using some combinatorial techniques, in this note, it is proved that if $\alpha\geq 0.28866$, then any digraph on $n$ vertices with minimum outdegree at least $\alpha n$ contains a directed cycle of length at most 4.

We prove that for every $\varepsilon > 0$ there exists $n_0=n_0(\varepsilon)$ such that every regular oriented graph on $n > n_0$ vertices and degree at least $(1/4 + \varepsilon)n$ has a Hamilton cycle. This establishes an approximate…

In a graph $G$, a subset of vertices $S \subseteq V(G)$ is said to be cyclable if there is a cycle containing the vertices in some order. $G$ is said to be $k$-cyclable if any subset of $k \geq 2$ vertices is cyclable. If any $k$…

We prove that there exists a function $f:\mathbb{N}\rightarrow \mathbb{R}$ such that every directed graph $G$ contains either $k$ directed odd cycles where every vertex of $G$ is contained in at most two of them, or a set of at most $f(k)$…

In 1963, Corr\'adi and Hajnal proved that for all $k\geq1$ and $n\geq3k$, every graph $G$ on $n$ vertices with minimum degree $\delta(G)\geq2k$ contains $k$ disjoint cycles. The bound $\delta(G) \geq 2k$ is sharp. Here we characterize those…

It is conjectured that every edge-colored complete graph $G$ on $n$ vertices satisfying $\Delta^{mon}(G)\leq n-3k+1$ contains $k$ vertex-disjoint properly edge-colored cycles. We confirm this conjecture for $k=2$, prove several additional…

Consider a graph with $n$ vertices where the shortest odd cycle is of length $>2k+1$. We revisit two known results about such graphs: (I) Such a graph is almost bipartite, in the sense that it can be made bipartite by removing from it…

A pseudo 2-factor of a graph is a spanning subgraph such that each component is $K_1$, $K_2$, or a cycle. This notion was introduced by Bekkai and Kouider in 2009, where they showed that every graph $G$ has a pseudo 2-factor with at most…

The classical Erd\H{o}s-P\'{o}sa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k+1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k)…

We conjecture that the balanced complete bipartite graph $K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil}$ contains more cycles than any other $n$-vertex triangle-free graph, and we make some progress toward proving this. We give equivalent…

We prove that for any $t\ge 3$ there exist constants $c>0$ and $n_0$ such that any $d$-regular $n$-vertex graph $G$ with $t\mid n\geq n_0$ and second largest eigenvalue in absolute value $\lambda$ satisfying $\lambda\le c d^{t}/n^{t-1}$…