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In this paper we study some open questions related to the smallest order$f({\cal C},\lnot {\cal H})$ of a 4-regular graph which has a connectivityproperty ${{\cal C}}$ but does not have a hamiltonian property ${\cal H}$. Inparticular, ${\cal C}$ is either connectivity, 2-connectivity or 1-toughnessand ${\cal H}$ is hamiltonicity, homogeneously traceability or traceability. Astandard theoretical approach to these questions had already been used in theliterature, but did not succeed in determining the exact value of $f()$. Herewe have chosen to use Integer Linear Programming and to encode the graphs thatwe are looking for as the binary solutions to a suitable set of linearinequalities. This way, there would exist a graph of order $n$ with certainproperties if and only if the corresponding ILP had a feasible solution, whichwe have determined through a branch-and-cut procedure. By using our approach,we have been able to compute $f({\cal C},\lnot {\cal H})$ for all the pairs ofconsidered properties with the exception of ${\cal C}=$1-toughness, ${{\calH}}=$traceability. Even in this last case, we have nonetheless significantlyreduced the interval $[LB, UB]$ in which $f({\cal C},\lnot {\cal H})$ was knownto lie. Finally, we have shown that for each $n \geq f({\cal C},\lnot {\calH})$ ($n \geq UB$ in the last case) there exists a 4-regular graph on $n$vertices which has property ${\cal C}$ but not property ${\cal H}$.

Proving hamiltonian properties in connected 4-regular graphs: an ILP-based approach