A plane version of the Fleischner theorem

Let $\cal G$ be a family of all $3$-regular $2$-connected plane multigraphs without loops. We prove the following plane version of the Fleischner theorem: Let $G$ be a graph in $\cal G$. For every $2$-factor $X$ of $G$ having $n$-components there exists a plane graph $J$ having a Hamilton cycle omitting all edges of $E(G)\backslash E(X)$ and such that $G \subseteq J \subset G^{2}$, $\Delta(J) \leqslant 5$ and $|E(J)|= |E(G)| + 2n -2$. Moreover, if $G$ is simple, then $J$ is simple too.Comment: 5 pages, 2 figure

A hamiltonian cycle in the square of a 2-connected graph in linear time

Fleischner’s theorem says that the square of every 2-connected graph contains a Hamiltonian cycle. We present a proof resulting in an O(|E|) algorithm for producing a Hamiltonian cycle in the square G2 of a 2-connected graph G = (V, E). The previous best was O(|V|2) by Lau in 1980. More generally, we get an O(|E|) algorithm for producing a Hamiltonian path between any two prescribed vertices, and we get an O(|V|2) algorithm for producing cycles C3, C4 , . . . , C | V | in G2 of lengths 3,4 , . . . , |V|, respectively