TY - JOUR
T1 - A new intrinsically knotted graph with 22 edges
AU - Kim, Hyoungjun
AU - Lee, Hwa Jeong
AU - Lee, Minjung
AU - Mattman, Thomas
AU - Oh, Seungsang
N1 - Funding Information:
The fifth author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. NRF-2014R1A2A1A11050999).
Publisher Copyright:
© 2017 Elsevier B.V.
PY - 2017/9/1
Y1 - 2017/9/1
N2 - 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, independently, Barsotti and Mattman) proved there are exactly 14 intrinsically knotted graphs with 21 edges by showing that H12 and C14 are the only triangle-free intrinsically knotted graphs of size 21. Our current goal is to find the complete set of intrinsically knotted graphs with 22 edges. To this end, using the main argument in [9], we seek triangle-free intrinsically knotted graphs. In this paper we present a new intrinsically knotted graph with 22 edges, called M11. We also show that there are exactly three triangle-free intrinsically knotted graphs of size 22 among graphs having at least two vertices with degree 5: cousins 94 and 110 of the E9+e family, and M11. Furthermore, there is no triangle-free intrinsically knotted graph with 22 edges that has a vertex with degree larger than 5.
AB - 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, independently, Barsotti and Mattman) proved there are exactly 14 intrinsically knotted graphs with 21 edges by showing that H12 and C14 are the only triangle-free intrinsically knotted graphs of size 21. Our current goal is to find the complete set of intrinsically knotted graphs with 22 edges. To this end, using the main argument in [9], we seek triangle-free intrinsically knotted graphs. In this paper we present a new intrinsically knotted graph with 22 edges, called M11. We also show that there are exactly three triangle-free intrinsically knotted graphs of size 22 among graphs having at least two vertices with degree 5: cousins 94 and 110 of the E9+e family, and M11. Furthermore, there is no triangle-free intrinsically knotted graph with 22 edges that has a vertex with degree larger than 5.
KW - Intrinsically knotted
KW - Spatial graph
UR - http://www.scopus.com/inward/record.url?scp=85021108205&partnerID=8YFLogxK
U2 - 10.1016/j.topol.2017.06.013
DO - 10.1016/j.topol.2017.06.013
M3 - Article
AN - SCOPUS:85021108205
SN - 0166-8641
VL - 228
SP - 303
EP - 317
JO - Topology and its Applications
JF - Topology and its Applications
ER -