Related papers: Semi-Degree Condition for Arbitrary $H$-Linked Ori…

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 cycle of length $t$ in a hypergraph is an alternating sequence $v_1,e_1,v_2\dots,v_t,e_t$ of distinct vertices $v_i$ and distinct edges $e_i$ so that $\{v_i,v_{i+1}\}\subseteq e_i$ (with $v_{t+1}:=v_1$). Let $\lambda K_n^h$ be the…

In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the…

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…

A famous result by R\"odl, Ruci\'nski, and Szemer\'edi guarantees a (tight) Hamilton cycle in $k$-uniform hypergraphs $H$ on $n$ vertices with minimum $(k-1)$-degree $\delta_{k-1}(H)\geq (1/2+o(1))n$, thereby extending Dirac's result from…

We show that for every $\eta>0$ every sufficiently large $n$-vertex oriented graph D of minimum semidegree exceeding $(1 + \eta) k/2$ contains every balanced antidirected tree with $k$ edges and bounded maximum degree, if $k \ge \eta n$. In…

We present a tight extremal threshold for the existence of Hamilton cycles in graphs with large minimum degree and without a large ``bipartite hole`` (two disjoint sets of vertices with no edges between them). This result extends Dirac's…

As an extension of the Brooks theorem, Catlin in 1979 showed that if $H$ is neither an odd cycle nor a complete graph with maximum degree $\Delta(H)$, then $H$ has a vertex $\Delta(H)$-coloring such that one of the color classes is a…

Let $D$ be a digraph on $p\geq 5$ vertices with minimum degree at least $p-1$ and with minimum semi-degree at least $p/2-1$. For $D$ (unless some extremal cases) we present a detailed proof of the following results [12]: (i) $D$ contains…

Let $\mathbb{Z}_q$ denote the cyclic group of order $q$. A $\mathbb{Z}_q$-edge-weighted $K_f$ is the complete graph $K_f$ equipped with a weight function $\omega : E(K_f) \to \mathbb{Z}_q$. A subdivision of a graph $H$ in a…

The cycle space $\mathcal{C}(G)$ of a graph $G$ is defined as the linear space spanned by all cycles in $G$. For an integer $k\ge 3$, let $\mathcal{C}_k (G)$ denote the subspace of $\mathcal{C}(G)$ generated by the cycles of length exactly…

Understanding how the cycles of a graph or digraph behave in general has always been an important point of graph theory. In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct…

We show that for any natural number $k \ge 1$, any oriented graph $D$ of minimum semidegree at least $(3k- 2)/4$ contains an antidirected path of length $k$. In fact, a slightly weaker condition on the semidegree sequence of $D$ suffices,…

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…

A graph $G$ contains another graph $H$ as an immersion if $H$ can be obtained from a subgraph of $G$ by splitting off edges and removing isolated vertices. There is an obvious necessary degree condition for the immersion containment: if $G$…

We first prove that for every vertex x of a 4-connected graph G there exists a subgraph H in G isomorphic to a subdivision of the complete graph K4 on four vertices such that G-V(H) is connected and contains x. This implies an affirmative…

Bedrossian characterized all pairs of forbidden subgraphs for a 2-connected graph to be Hamiltonian. Instead of forbidding some induced subgraphs, we relax the conditions for graphs to be Hamiltonian by restricting Ore- and Fan-type degree…

A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh and Staden proved that among all graphs with minimum degree $d$, $K_{d+1}$ minimises the number of…

One way to certify that a graph does not contain an induced cycle of length six is to provide a partition of its vertex set into (i) a stable set, and (ii) a graph containing no stable set of size three and no induced matching of size two.…

We define a $(V_1, V_2, \ldots, V_k)$-partition for a given graph $H$ and graphical properties $P_1, P_2, \ldots, P_k$ as a partition where each $V_i$ induces a subgraph of $H$ with property $P_i$. Matamala (2007) extended this result by…