2023/12/11 by Khandhawit, Tirasan, Pongtanapaisan, Puttipong, Wasun, Athibadee
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2312.06603
Two isomorphic graphs can have inequivalent spatial embeddings in 3-space. In this way, an isomorphism class of graphs contains many spatial graph types. A common way to measure the complexity of a spatial graph type is to count the minimum number of straight sticks needed for its construction in 3-space. In this paper, we give estimates of this quantity by enumerating stick diagrams in a plane. In particular, we compute the planar stick indices of knotted graphs with low crossing numbers. We also show that if a bouquet graph or a theta-curve has the property that its proper subgraphs are all trivial, then the planar stick index must be at least seven.