Related papers: Small Complete Minors Above the Extremal Edge Dens…

Goodman proved that the sum of the number of triangles in a graph on $n$ nodes and its complement is at least $n^3/24$; in other words, this sum is minimized, asymptotically, by a random graph with edge density $1/2$. Erd\H{o}s conjectured…

A good edge-labeling of a graph [Ara\'ujo, Cohen, Giroire, Havet, Discrete Appl. Math., forthcoming] is an assignment of numbers to the edges such that for no pair of vertices, there exist two non-decreasing paths. In this paper, we study…

A graph $H$ is an induced minor of a graph $G$ if $H$ can be obtained from $G$ by vertex deletions and edge contractions. We show that there is a function $f(k, d) = O(k^{10} + 2^{d^5})$ so that if a graph has treewidth at least $f(k, d)$…

In 1975 Bollob\'{a}s, Erd\H{o}s, and Szemer\'{e}di asked what minimum degree guarantees an octahedral subgraph $K_3(2)$ in any tripartite graph $G$ with $n$ vertices in each vertex class. We show that $\delta(G)\geq n+2n^{\frac{5}{6}}$…

A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one another edge. In this work we prove that each 1-planar graph of minimum degree at least $3$ contains an edge with degrees of its endvertices of…

Archdeacon (1987) proved that graphs embeddable on a fixed surface can be $3$-coloured so that each colour class induces a subgraph of bounded maximum degree. Edwards, Kang, Kim, Oum and Seymour (2015) proved that graphs with no…

An important result of Koml\'os [Tiling Tur\'an theorems, Combinatorica, 2000] yields the asymptotically exact minimum degree threshold that ensures a graph $G$ contains an $H$-tiling covering an $x$th proportion of the vertices of $G$ (for…

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log…

A celebrated theorem of Stiebitz asserts that any graph with minimum degree at least $s+t+1$ can be partitioned into two parts which induce two subgraphs with minimum degree at least $s$ and $t$, respectively. This resolved a conjecture of…

We provide a short and self-contained proof of the classical result of Kostochka and of Thomason, ensuring that every graph of average degree $d$ has a complete minor of order $d/\sqrt{\log d}$.

Hadwiger's conjecture, among the most famous open problems in graph theory, states that every graph that does not contain $K_t$ as a minor is properly $(t-1)$-colorable. The purpose of this work is to demonstrate that a natural extension of…

Let $H$ be a $k$-uniform hypergraph on $n$ vertices where $n$ is a sufficiently large integer not divisible by $k$. We prove that if the minimum $(k-1)$-degree of $H$ is at least $\lfloor n/k \rfloor$, then $H$ contains a matching with…

A tree $T$ in an edge-colored graph is a {\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be a fixed integer with $2\le k\le n$. For a vertex subset $S \subseteq…

Hadwiger's Conjecture asserts that every $K_t$-minor-free graph has a proper $(t-1)$-colouring. We relax the conclusion in Hadwiger's Conjecture via improper colourings. We prove that every $K_t$-minor-free graph is $(2t-2)$-colourable with…

We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of $k$-edge stars in a graph with density $x \in…

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…

For a graph $G$, let $c_k(G)$ be the number of spanning trees of $G$ with maximum degree at most $k$. For $k \ge 3$, it is proved that every connected $n$-vertex $r$-regular graph $G$ with $r \ge \frac{n}{k+1}$ satisfies $$ c_k(G)^{1/n} \ge…

We investigate Hadwiger's conjecture for graphs with no stable set of size 3. Such a graph on at least 2t-1 vertices is not t-1 colorable, so is conjectured to have a $K_t$ minor. There is a strengthening of Hadwiger's conjecture in this…

For a positive integer $k$, a graph is $k$-knitted if for each $k$-subset $S$ of vertices, and every partition of $S$ into disjoint parts $S_1, \ldots, S_t$ for some $t\ge 1$, one can find disjoint connected subgraphs $C_1, \ldots, C_t$…

We characterize classes of graphs closed under taking vertex-minors and having no $P_n$ and no disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ for some $n$. Our characterization is described in terms of a tree of radius $2$…