3 problems

A prime labeling of a graph with $n$ vertices assigns its vertices distinct integers from $\{1,2,\ldots,n\}$ so that adjacent vertices receive relatively prime labels; a graph is p…

For odd integers $n$ and $k$ with $n\ge 3$, consider the cycle $C_n$ and the consecutive label set $\{k,k+1,\ldots,k+n-1\}$. A prime labeling assigns these labels bijectively to th…

Let $G=C_{n_1}\cup\cdots\cup C_{n_m}$ be a $2$-regular graph, where each $n_i\ge 3$, and let a graph be prime when it admits a labeling by distinct integers from…