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paths on a graph that visit each vertex exactly once

3 votes
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Hamiltonian path in total graph

The edges incident to $v \in V$ form a clique in $T(G)$, so the Hamiltonian path $v_1, v_2, v_3, \ldots, v_n$ in $G$ can be lifted into a Hamiltonian path in $T(G)$ as follows: Insert the edges to ge …
Peter Taylor's user avatar
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3 votes
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Path of length $n$ but no Hamilton cycle

Assuming you mean a graph with a Hamiltonian path but no Hamiltonian cycle, The Petersen graph is a standard example. Any pendant-free graph with a Hamiltonian path and a bridge is an easy example; e …
Peter Taylor's user avatar
  • 7,226
3 votes
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How to construct a hamilton-connected cubic graph? Is it possible?

Take a cycle of $2m$ elements, label the vertices around the cycle from $0$ to $2m-1$, and add diagonals $1 \to (2m-1)$, $2 \to (2m-2)$, $\cdots$, $(m-1) \to (m+1)$, and special case $0 \to m$. By sym …
Peter Taylor's user avatar
  • 7,226