Related papers: Intrinsic linking and knotting are arbitrarily com…
We list more than 200 new examples of minor minimal intrinsically knotted graphs and describe many more that are intrinsically knotted and likely minor minimal.
(1) We show that if a presentation of the trivial group is "hard to trivialize", in the sense that lots of Tietze moves are necessary to transform it into the trivial presentation, then the associated presentation complex (which is a…
The analogue of Hadwiger's conjecture for the immersion relation states that every graph $G$ contains an immersion of $K_{\chi(G)}$. For graphs with independence number 2, this is equivalent to stating that every such $n$-vertex graph…
Let L be an oriented (d+1)-component link in the 3-sphere, and let L(q) be the d-component link in a homology 3-sphere that results from performing 1/q-surgery on the last component. Results about the Alexander polynomial and twisted…
Given any oriented link diagram, two types of new knot invariants are constructed. They satisfy some generalized skein relations. The coefficients of each invariant is from a commutative ring. Homomorphisms and representations of those…
We use a randomised embedding method to prove that for all \alpha>0 any sufficiently large oriented graph G with minimum in-degree and out-degree \delta^+(G),\delta^-(G)\geq (3/8+\alpha)|G| contains every possible orientation of a Hamilton…
It is shown that if a link in 3-space bounds a proper oriented surface (without closed component) in the upper half 4-space, then the link bounds a proper oriented ribbon surface in the upper half 4-space which is a renewal embedding of the…
We prove that if $H$ is a subgraph of a complete multipartite graph $G$, then $H$ contains a connected component $H'$ satisfying $|E(H')||E(G)|\geq |E(H)|^2$. We use this to prove that every three-coloring of the edges of a complete graph…
We consider an algebra of (classical or virtual) tangles over an ordered circuit operad and introduce Conway-type invariants of tangles which respect this algebraic structure. The resulting invariants contain both the coefficients of the…
Polyak showed that any Milnor's $\overline{\mu}$-invariant of length 3 can be represented as a combination of Conway polynomials of knots obtained by certain band sum of the link components. On the other hand, Habegger and Lin showed that…
The first and last named authors have demonstrated the existence of knots for which every integral slope is non-characterizing. In this short note, we extend this result in two ways. There exists a knot that shares for every integer n the…
We prove a version of the Loebl-Komlos-Sos Conjecture for dense graphs. For each q>0 there exists a number $n_0\in \mathbb{N}$ such that for any n>n_0 and k>qn the following holds: if G be a graph of order n with at least n/2 vertices of…
Shrub-depth and rank-depth are related graph parameters that are dense analogs of tree-depth. We prove that for every positive integer $t$, every graph of sufficiently large rank-depth contains a pivot-minor isomorphic to a path on $t$…
The linking integral is an invariant of the link-type of two manifolds immersed in a Euclidean space. It is shown that the ordinary Gauss integral in three dimensions may be simplified to a winding number integral in two dimensions. This…
The connected graph of degree sequence 3,3,3,1,1,1 is called a net, and the vertices of degree 1 in a net is called its endvertices. Broersma conjectured in 1993 that a 2-connected graph G with no induced K_{1,3} is hamiltonian if every…
Consider a graph $G$ on $n$ vertices with $\alpha \binom{n}{2}$ edges which does not contain an induced $K_{2, t}$ ($t \geqslant 2$). How large does $\alpha$ have to be to ensure that $G$ contains, say, a large clique or some fixed subgraph…
An inscribed knot is formed by polygonally connecting points lying on a knot $\gamma$ in parametric order, then closing the path by connecting the first and final points. The stick-knot number of a knot type K is the minimum number of line…
The analogue of Hadwiger's conjecture for the immersion order states that every graph $G$ contains $K_{\chi (G)}$ as an immersion. If true, it would imply that every graph with $n$ vertices and independence number $\alpha$ contains…
This is the second of a series of four papers in which we prove the following relaxation of the Loebl-Komlos--Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at…
We prove that for every graph $G$ with a sufficiently large complete bipartite induced minor, either $G$ has an induced minor isomorphic to a large wall, or $G$ contains a large constellation; that is, a complete bipartite induced minor…