Intrinsically triple linked complete graphs

An intrinsic nontriviality of graphs

“…Since F 0 is also a subgraph of G , we have that G is intrinsically nonfree. Finally we show that (1) implies (3). Assume that G is not intrinsically linked.…”

“…Conway and Gordon also showed that K 7 is intrinsically knotted, namely every spatial embedding f of G contains a nontrivial knot. For a positive integer n, Flapan, Foisy, Naimi and Pommersheim [4] showed that there exists an intrinsically n-linked graph G , namely every spatial embedding f of G contains a nonsplittable n-component link (see also Flapan-Naimi-Pommersheim [3] and Bowlin-Foisy [1] for the case of n D 3). Note that these results paid attention to only constituent knots and links of spatial graphs.…”

INTRINSICALLY n-LINKED COMPLETE BIPARTITE GRAPHS

“…First based on the number of disjoint simple closed curves needed in K m we obtain a lower bound of 3n. This lower bound is realized in the n = 2 case, but no longer for the n = 3 case, where m = 10 [4]. It follows from [3] that K 7n−6 is intrinsically n-linked.…”

“…A link L is split if there is an embedding of a 2-sphere F in R 3 L such that each component of R 3 F contains at least one component of L. A graph G is intrinsically n-linked if every embedding of G into R 3 contains a non-split n-component link. Flapan, Naimi, and Pommersheim investigate intrinsically triple linked graphs in [4]. They proved that K 10 is the smallest complete graph to be intrinsically triple linked.…”

A newly recognized intrinsically knotted graph

“…The case when the linked five cycle is ðv; 5; 6; 7; 12Þ is similarly difficult. To deal with these cases, we make use of the following elementary observation from homology theory, also used in [3], which we state without proof:…”