Related papers: Counting Links and Knots in Complete Graphs
We find the minimal number of links in an embedding of any complete $k$-partite graph on 7 vertices (including $K_7$, which has at least 21 links). We give either exact values or upper and lower bounds for the minimal number of links for…
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,…
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…
Minimum braids are a complete invariant of knots and links. This paper defines minimum braids, describes how they can be generated, presents tables for knots up to ten crossings and oriented links up to nine crossings, and uses minimum…
The splitting number of a link is the minimum number of crossing changes between distinct components that is required to convert the link into a split link. We provide a bound on the splitting number in terms of the four-genus of related…
It is well-known that 1-planar graphs have minimum degree at most 7, and not hard to see that some 1-planar graphs have minimum degree exactly 7. In this note we show that any such 1-planar graph has at least 24 vertices, and this is tight.
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 $C_1, \ldots, C_t$…
A graph is k-linked if any k disjoint vertex-pairs can be joined by k disjoint paths. We improve a lower bound on the linkedness of polytopes slightly, which results in exact values for the minimal linkedness of 7-, 10- and 13-dimensional…
Connected Vertex Cover is one of the classical problems of computer science, already mentioned in the monograph of Garey and Johnson. Although the optimization and decision variants of finding connected vertex covers of minimum size or…
We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that…
Algorithm of construction of all knots, links with given number of crosses on diagram of knot, link is offered. This algorithm is based on simple proposition, that there is a representation of knot (link) as closure of braid with n threads…
We study the minimum rank of a (simple, undirected) graph, which is the minimum rank among all matrices in a space determined by the graph. We determine the exact set of graphs on eight vertices for which the nullity of a minimum rank…
The lattice stick number of knots is defined to be the minimal number of straight sticks in the cubic lattice required to construct a lattice stick presentation of the knot. We similarly define the lattice stick number $s_{L}(G)$ of spatial…
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…
The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering…
Answering a question of Erd\H{o}s and Ne\v{s}et\v{r}il, we show that the maximum number of inclusion-wise minimal vertex cuts in a graph on $n$ vertices is at most $1.8899^n$ for large enough $n$.
Given a finite, simple graph $G$, the $k$-component order edge connectivity of $G$ is the minimum number of edges whose removal results in a subgraph for which every component has order at most $k-1$. In general, determining the…
Knots and links have been considered to be useful models for structural analysis of molecular chains such as DNA and proteins. One quantity that we are interested on molecular links is the minimum number of monomers necessary to realize…
A connected graph $G$ with at least two vertices is matching covered if each of its edges lies in a perfect matching. A matching covered graph is minimal if the removal of any edge results in a graph that is no longer matching covered. An…
We study the minimum ribbonlength for immersed planar ribbon knots and links. Our approach is to embed the space of such knots and links into a larger more tractable space of disk diagrams. When length minimisers in disk diagram space are…