WonHyuk Choi | Searcharxiv

arXiv2018

Boltzmann Enhancements of Biquasile Counting Invariants

In this paper, we build on the biquasiles and dual graph diagrams introduced in arXiv:1610.06969. We introduce \textit{biquasile Boltzmann weights} that enhance the previous knot coloring invariant defined in terms of finite biquasiles and provide examples differentiating links with the same counting invariant, demonstrating that the enhancement is proper. We identify conditions for a linear function $ϕ:\mathbb{Z}_n[X^3]\to\mathbb{Z}_n$ to be a Boltzmann weight for an Alexander biquasile $X$.

math.GT↗

arXiv2015

$(r)$-Pancyclic, $(r)$-Bipancyclic and Oddly $(r)$-Bipancyclic Graphs

A graph with $v$ vertices is $(r)$-pancyclic if it contains precisely $r$ cycles of every length from 3 to $v$. A bipartite graph with even number of vertices $v$ is said to be $(r)$-bipancyclic if it contains precisely $r$ cycles of each even length from 4 to $v$. A bipartite graph with odd number of vertices $v$ and minimum degree at least 2 is said to be oddly $(r)$-bipancyclic if it contains precisely $r$ cycles of each even length from 4 to $v-1$. In this paper, using computer search, we classify all $(r)$-pancyclic and $(r)$-bipancyclic graphs with $v$ vertices and at most $v+5$ edges. We also classify all oddly $(r)$-bipancyclic graphs with $v$ vertices and at most $v+4$ edges.

math.CO↗