Related papers: When all directed cycles have length three

We prove that the directed 3-torus D_3(m), or equivalently the Cartesian product of three directed m-cycles, admits a decomposition into three arc-disjoint directed Hamilton cycles for every integer m >= 3. The proof reduces Hamiltonicity…

The circumference of a graph is the length of its longest cycles. Jackson established a conjecture of Bondy by showing that the circumference of a 3-connected cubic graph of order $n$ is $\Omega(n^{0.694})$. Bilinski {\it et al.} improved…

Let $k$ and $r$ be two integers with $k \ge 2$ and $k\ge r \ge 1$. In this paper we show that (1) if a strongly connected digraph $D$ contains no directed cycle of length $1$ modulo $k$, then $D$ is $k$-colorable; and (2) if a digraph $D$…

Let $G$ be a graph of radius $r$ and diameter $d$ with $d\leq 2r-2$. We give a new proof that $G$ contains a cycle of length at least $4r-2d$, i.e. for its circumference it holds $c(G)\geq 4r-2d$.

The three subgraphs of a connected graph induced by the center, annulus and periphery are called its metric subgraphs. The main results are as follows. (1) There exists a graph of order $n$ whose metric subgraphs are all paths if and only…

Let $D$ be a digraph of order $p\geq5$ with minimum degree at least $p-1$ and with minimum semi-degree at least $p/2-1$. In his excellent and renowned paper, ``Long Cycles in Digraphs" (Proc. London Mathematical Society (3), 42 (1981),…

Let $ H $ be a multi-digraph on $ h $ vertices with $ q $ arcs. An \textbf{$H$-subdivision} in a digraph $D$ is a subdigraph obtained by replacing every arc $uv$ of $H$ with a path from $u$ to $v$ in $D$ such that these paths are pairwise…

We show that $3$-graphs on $n$ vertices whose codegree is at least $(2/3 + o(1))n$ can be decomposed into tight cycles and admit Euler tours, subject to the trivial necessary divisibility conditions. We also provide a construction showing…

The Oberwolfach problem asks for a $2$-factorization of the complete graph in which each $2$-factor is isomorphic to a specific factor $F$. Recently, this problem has been extended to directed graphs. In this case, the directed Oberwolfach…

Let $G$ be a graph of radius $r$ and diameter $d$ with $d\leq 2r-2$. We show that $G$ contains a cycle of length at least $4r-2d$, i.e. for its circumference it holds $c(G)\geq 4r-2d$. Moreover, for all positive integers $r$ and $d$ with…

A well known theorem in graph theory states that every graph $G$ on $n$ vertices and minimum degree at least $d$ contains a path of length at least $d$, and if $G$ is connected and $n\ge 2d+1$ then $G$ contains a path of length at least…

Let $D$ be a strongly connected directed graph of order $n\geq 4$. In \cite{[14]} (J. of Graph Theory, Vol.16, No. 5, 51-59, 1992) Y. Manoussakis proved the following theorem: Suppose that $D$ satisfies the following condition for every…

A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. The problem of finding rainbow subgraphs goes back to the work of Euler on transversals in Latin squares and was extensively studied since then.…

Let G be a planar triangle-free graph and let C be a cycle in G of length at most 8. We characterize all situations where a 3-coloring of C does not extend to a proper 3-coloring of the whole graph.

The Hamiltonian cycle problem in digraph is mapped into a matching cover bipartite graph. Based on this mapping, it is proved that determining existence a Hamiltonian cycle in graph is $O(n^3)$.

Let $G$ be a graph, and let $f_G$ be the sum of $(-1)^{|A|}$, over all stable sets $A$. If $G$ is a cycle with length divisible by three, then $f_G= \pm 2$. Motivated by topological considerations, G. Kalai and R. Meshulam made the…

Let $D=(V(D), A(D))$ be a digraph of order $n$ and let $S\subseteq V(D)$ with $2\leq |S|\leq n$. A directed cycle $C$ of $D$ is called a directed $S$-Steiner cycle (or, an $S$-cycle for short) if $S\subseteq V(C)$. Steiner cycles have…

A Berge cycle of length $\ell$ in a hypergraph is an alternating sequence of $\ell$ distinct vertices and $\ell$ distinct edges $v_1,e_1,v_2, \ldots, v_\ell, e_{\ell}$ such that $\{v_i, v_{i+1}\} \subseteq e_i$ for all $i$, with indices…

We prove a 1985 conjecture of Gy\'arf\'as that for all $k,\ell$, every graph with sufficiently large chromatic number contains either a complete subgraph with $k$ vertices or an induced cycle of length at least $\ell$.

We prove that a connected graph contains a circuit---a closed walk that repeats no edges---through any $k$ prescribed edges if and only if it contains no odd cut of size at most $k$.