Related papers: A note on complete subdivisions in digraphs of lar…

Hadwiger's conjecture asserts that if a simple graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $t$ stable sets. This is still open, but we prove under the same hypotheses that $V(G)$ can be partitioned…

Generalizing well-known results of Erd\H{o}s and Lov\'asz, we show that every graph $G$ contains a spanning $k$-partite subgraph $H$ with $\lambda{}(H)\geq \lceil{}\frac{k-1}{k}\lambda{}(G)\rceil$, where $\lambda{}(G)$ is the…

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 $G$…

Given a (di)graph $H$, we say that a (di)graph $H^\prime$ is an $H$-subdivision if $H^\prime$ is obtained from $H$ by replacing one or more edges with internally vertex-disjoint path(s). Pavez-Sign\'{e} conjectured that for every…

A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint copies of the clique $K_r$ in $G$ covering every vertex of $G$. The famous Hajnal--Szemer\'edi theorem determines the minimum degree threshold for forcing a perfect…

Let $D=(V,A)$ be an acyclic digraph. For $x\in V$ define $e_{_{D}}(x)$ to be the difference of the indegree and the outdegree of $x$. An acyclic ordering of the vertices of $D$ is a one-to-one map $g: V \rightarrow [1,|V|] $ that has the…

Seymour's second neighbourhood conjecture asserts that every oriented graph has a vertex whose second out-neighbourhood is at least as large as its out-neighbourhood. In this paper, we prove that the conjecture holds for quasi-transitive…

Dean conjectured three decades ago that every graph with minimum degree at least $k\ge 3$ contains a cycle whose length is divisible by $k$. While the conjecture has been verified for $k\in \{3,4\}$, it remains open for $k\ge 5$. A weaker…

Let $A$ and $B$ be sets of vertices in a graph $G$. Menger's theorem states that for every positive integer $k$, either there exists a collection of $k$ vertex-disjoint paths between $A$ and $B$, or $A$ can be separated from $B$ by a set of…

There are many intriguing questions in extremal graph theory that are well-understood in the undirected setting and yet remain elusive for digraphs. A natural instance of such a problem was recently studied by Hons, Klimo\v{s}ov\'{a},…

Let $G$ be an $r$-uniform hypergraph on $n$ vertices such that all but at most $\varepsilon \binom{n}{\ell}$ $\ell$-subsets of vertices have degree at least $p \binom{n-\ell}{r-\ell}$. We show that $G$ contains a large subgraph with high…

Let $G$ be a finite, simple, and undirected graph of order $n$ and average degree $d$. Up to terms of smaller order, we characterize the minimal intervals $I$ containing $d$ that are guaranteed to contain some vertex degree. In particular,…

Thomassen, in 1983, conjectured that for a positive integer $k$, every $2$-connected non-bipartite graph of minimum degree at least $k + 1$ contains cycles of all lengths modulo $k$. In this paper, we settle this conjecture affirmatively.

In this article we have derived the minimum order of an odd regular graph such that the graph has no matching. We have observed that how it is different from the case of even regular graphs. We have checked the consistency of the derived…

We suggest two related conjectures dealing with the existence of spanning irregular subgraphs of graphs. The first asserts that any $d$-regular graph on $n$ vertices contains a spanning subgraph in which the number of vertices of each…

Given positive integers k\geq 3 and r where k/2 \leq r \leq k-1, we give a minimum r-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko who gave an…

Let $k$, $m$ and $r$ be three integers such that $2\leq k\leq m\leq r$. Let $G$ be a $2r$-regular, $2m$-edge-connected graph of odd order. We obtain some sufficient conditions, which guarantee $G-v$ contains a $k$-factor for all $v\in…

We prove that for each $D\ge 2$ there exists $c>0$ such that whenever $b\le c\big(\tfrac{n}{\log n}\big)^{1/D}$, in the $(1:b)$ Maker-Breaker game played on $E(K_n)$, Maker has a strategy to guarantee claiming a graph $G$ containing copies…

Let $D=(V,A)$ be a directed graph of order $n\geq 6$. Let $W$ be a subset of $V$ with $|W|\geq 6$. Suppose that every vertex of $W$ has degree at least $(3n-3)/2$ in $D$. Then for any integer partition $|W|=n_1+n_2$ with $n_1\geq 3$ and…

A graph $G$ is $k$-degenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree $k$ or less. We prove that every connected planar graph with average degree $d \ge 2$ has a 4-degenerate induced…