[Paper Review] Intrinsic knotting and linking of almost complete partite graphs

Thomas W. Mattman, Ryan Ottman+1 more|ArXiv.org|Dec 9, 2003

Advanced Graph Theory Research8 references3 citations

TL;DR

This paper classifies intrinsic knotting and linking in 0-, 1-, and 2-deficient complete and complete partite graphs, and fully characterizes all intrinsically knotted graphs on 8 vertices. It verifies Adams' conjecture—that removing any vertex from an intrinsically knotted graph yields an intrinsically linked graph—for these families, confirming the conjecture holds for all such graphs up to 8 vertices, despite a known counterexample at 13 deficiency.

ABSTRACT

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 verify a conjecture presented in Adams' "The Knot Book": If a vertex is removed from an intrinsically knotted graph, one obtains an intrinsically linked graph.

Motivation & Objective

• To classify intrinsic knotting and linking in graphs that are 0, 1, or 2 edges short of being complete or complete partite graphs.
• To determine whether Adams' conjecture—that vertex removal from an intrinsically knotted graph yields an intrinsically linked graph—holds for these families.
• To fully enumerate and characterize all intrinsically knotted graphs on 8 vertices.
• To investigate whether graphs with more than 5n−15 edges can be non-intrinsically knotted, extending a question of Sachs to the context of intrinsic knotting.

Proposed method

• Systematically analyze 0-, 1-, and 2-deficient complete and complete partite graphs using minor theory and graph operations such as triangle-Y exchanges.
• Apply known minor-minimal intrinsically knotted graphs (e.g., from $K_7$ and $K_{3,3,1,1}$) as reference points for classification.
• Use edge deletion and vertex removal operations to test the intrinsic linking and knotting properties of subgraphs.
• Leverage the fact that $K_7$ and $H_8$ are minor-minimal intrinsically knotted to identify knotted subgraphs in $K_8$ with up to 8 edges removed.
• Construct complementary graphs for $K_8 - ke$ to identify knotted configurations, particularly for $k = 5,6,7$.
• Verify Adams’ conjecture by checking that vertex removal from each knotted graph yields an intrinsically linked minor or subgraph.

Experimental results

Research questions

• RQ1 Which 0-, 1-, or 2-deficient complete or complete partite graphs are intrinsically knotted or linked?
• RQ2 Does Adams' conjecture—that removing a vertex from an intrinsically knotted graph results in an intrinsically linked graph—hold for all such 0-, 1-, and 2-deficient graphs?
• RQ3 How many intrinsically knotted graphs exist on 8 vertices, and do they all satisfy Adams’ conjecture?
• RQ4 Is there a graph on $n$ vertices that is not intrinsically knotted but has more than $5n - 15$ edges, for $n > 8$?
• RQ5 Can the bound $5n - 15$ edges for non-intrinsically knotted graphs be extended beyond $n = 8$?

Key findings

• There are exactly twenty intrinsically knotted graphs on 8 vertices, all of which satisfy Adams’ conjecture.
• All 0-, 1-, and 2-deficient complete and complete partite graphs that are intrinsically knotted are also intrinsically linked upon vertex removal, confirming Adams’ conjecture for these families.
• No new minor-minimal intrinsically knotted graphs are found among 0-, 1-, or 2-deficient graphs or among 8-vertex graphs beyond the known examples from $K_7$ and $K_{3,3,1,1}$.
• The graph $H_8$ and $K_7$ with an additional vertex are both intrinsically knotted and minor-minimal, and vertex removal from either yields an intrinsically linked graph.
• For $5 \leq n \leq 8$, any graph with more than $5n - 15$ edges is intrinsically knotted, suggesting a potential threshold for intrinsic knotting.
• All $K_8 - ke$ graphs with $k \geq 8$ edges removed are not intrinsically knotted, confirming that the 20 knotted graphs on 8 vertices are the complete set.