There are at least two graphs associated with H. Walther.
The graph on 25 vertices illustrated above which is a variant of Tutte's fragment is depicted in Problem 2.10 of Harary (1994, p. 24; cf. Pemmaraju and Skiena(2003, p. 23). This graph is implemented in the Wolfram Language as GraphData["WaltherGraph25"].
A cubic nonhamiltonian graph on 162 vertices due Walther (1965) appears in Grünbaum (2003, Fig. 17.1.9, p. 366). It provides a counterexample to a conjecture of Hunter (1962) that all cyclically 5-connected polyhedral graphs contain a Hamiltonian cycle (Grünbaum 2003, p. 365). This graph will is implemented in the Wolfram Language as GraphData["WaltherGraph162"].