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Abstract

A graph is intrinsically knotted (IK) if for every embedding of the graph there exists a knotted cycle. Let G be a multipartite graph, and form the multipartite graph G’ by increasing the number of vertices in each of the parts except one and then deleting an edge. We show that if G is IK, then the resulting graph G’ is also IK. We use this idea to describe large families of IK multipartite graphs. In particular we use the fact that K5,5\2e is IK to show that a bipartite graph with 10 or more vertices (respectively 12 or more vertices) with exactly 5 (resp. 6) in one part and E(G) <= 4V(G)-17 (resp. E(G) <=5V(G)-27) is IK. Our method can’t be improved since we also show that K5,5\3e is not IK in general.

Author Bio

Chloe Collins is a senior at Portland State University, majoring in mathematics. This past summer she participated in a REU at CSU Chico under the direction of Dr. Thomas Mattman, they studied knots and their relationship to graphs. She plans to get her masters degree after graduating from Portland.

Ryan Hake is a junior majoring in mathematics at CSU Chico. This past summer he participated in a REU at CSU Chico under the direction of Dr. Thomas Mattman, they studied knots and their relationship to graphs. He plans to go on to receive a PhD. in mathematics following graduation in Spring of 2008.

Cara Petonic is a senior at Bryn Mawr, majoring in mathematics and dance. This past summer she participated in a REU at CSU Chico under the direction of Dr. Thomas Mattman, they studied knots and their relationship to graphs. She plans to work for an investment banking firm, following graduation from Bryn Mawr.

Laura Sardagna teaches mathematics at Academy of the Pacific in Honolulu, Hawaii.

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