Genus Ranges of 4-Regular Rigid Vertex Graphs
  We introduce a notion of genus range as a set of values of genera over all surfaces into which a graph is embedded cellularly, and we study the genus ranges of a special family of four-regular graphs with rigid vertices that has been used in modeling homologous DNA recombination. We show that the genus ranges are sets of consecutive integers. For any positive integer $n$, there are graphs with $2n $ vertices that have genus range ${m,m+1,...,m'}$ for all $0\le m<m'\le n$, and there are graphs with $2n-1$ vertices with genus range ${m,m+1,...,m'}$ for all $0\le m<m' <n$ or $0<m<m'\le n$. Further, we show that for every $n$ there is $k<n$ such that ${h}$ is a genus range for graphs with $2n-1$ and $2n$ vertices for all $h\le k$. It is also shown that for every $n$, there is a graph with $2n$ vertices with genus range ${0,1,...,n}$, but there is no such a graph with $2n-1$ vertices. 
Dorothy Buck, Egor Dolzhenko, Natasha Jonoska, Masahico Saito, Karin   Valencia
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