Related papers: Maximal knotless graphs
A maximally linkless graph is a graph that can be embedded in $\mathbb{R}^3$ without any links, but cannot be embedded in such a way if any other edge is added to the graph. Recently, a family of maximally linkless graphs was found with…
We construct a family of maximal linklessly embeddable graphs on $n$ vertices and $3n-5$ edges for all $n\ge 10$, and another family on $n$ vertices and $m< \frac{25n}{12}-\frac{1}{4}$ edges for all $n\ge 13$. The latter significantly…
A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and,…
A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael, and, independently, Mattman showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim,…
We classify all the maximal linklessly embeddable graphs of order 12 and show that their complements are all intrinsically knotted. We derive results about the connected domination numbers of a graph and its complement. We provide an answer…
We introduce new sufficient conditions for intrinsic knotting and linking. A graph on n vertices with at least 4n-9 edges is intrinsically linked. A graph on n vertices with at least 5n-14 edges is intrinsically knotted. We also classify…
We classify graphs that are 0, 1, or 2 edges short of being complete partite graphs with respect to intrinsic linking and intrinsic knotting. In addition, we classify intrinsic knotting of graphs on 8 vertices. For graphs in these families,…
A graph is intrinsically knotted if every embedding contains a nontrivially knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that there are exactly 14 intrinsically knotted graphs with 21 edges, in…
A connected graph $G$ with a perfect matching is said to be $k$-extendable for integers $k$, $1 \leq k\leq \frac{|V(G)|}{2}-1$, if any matching in $G$ of size $k$ is contained in a perfect matching of $G$. A $k$-extendable graph is minimal…
The complete bipartite graph $K_{1,3}$ is called a claw. The properties of claw-free graphs have attracted considerable attention, with research on claw-free planar graphs tracing back to Plummer's work in 1989. In this paper, we extend…
A graph is 2-apex if it is planar after the deletion of at most two vertices. Such graphs are not intrinsically knotted, IK. We investigate the converse, does not IK imply 2-apex? We determine the simplest possible counterexample, a graph…
It has been conjectured that for every claw-free graph $G$ the choice number of $G$ is equal to its chromatic number. We focus on the special case of this conjecture where $G$ is perfect. Claw-free perfect graphs can be decomposed via…
This article provides sharp bounds for the maximum number of edges possible in a simple graph with restricted values of two of the three parameters, namely, maxi- mum matching size, independence number and maximum degree. We also construct…
We present evidence in support of a conjecture that a bipartite graph with at least five vertices in each part and |E(G)| \geq 4 |V(G)| - 17 is intrinsically knotted. We prove the conjecture for graphs that have exactly five or exactly six…
We say that a graph is intrinsically knotted or completely 3-linked if every embedding of the graph into the 3-sphere contains a nontrivial knot or a 3-component link any of whose 2-component sublink is nonsplittable. We show that a graph…
A graph is intrinsically knotted if every embedding contains a knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that the KS graphs, $K_7$ and the 13 graphs obtained from $K_7$ by $\nabla Y$ moves, are…
A matching covered graph $G$ is minimal if for each edge $e$ of $G$, $G-e$ is not matching covered. An edge $e$ of a matching covered graph $G$ is removable if $G-e$ is also matching covered. Thus a matching covered graph is minimal if and…
A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k< n$, if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph is minimal if for every edge, the deletion of…
A graph is equimatchable if all of its maximal matchings have the same size. A graph is claw-free if it does not have a claw as an induced subgraph. In this paper, we provide, to the best of our knowledge, the first characterization of…
We present four models for a random graph and show that, in each case, the probability that a graph is intrinsically knotted goes to one as the number of vertices increases. We also argue that, for $k \geq 18$, most graphs of order $k$ are…