Traceability of locally hamiltonian and locally traceable graphs

Johan de Wet (1) , Susan van Aardt (1)

1 UNISA - University of South Africa

If $\mathcal{P}$ is a given graph property, we say that a graph $G$ is locally $\mathcal{P}$ if $\langle N(v) \rangle$ has property $\mathcal{P}$ for every $v \in V(G)$ where $\langle N(v) \rangle$ is the induced graph on the open neighbourhood of the vertex $v$. Pareek and Skupien (C. M. Pareek and Z. Skupien , On the smallest non-Hamiltonian locally Hamiltonian graph, J. Univ. Kuwait (Sci.), 10:9 - 17, 1983) posed the following two questions. Question 1 Is 9 the smallest order of a connected nontraceable locally traceable graph? Question 2 Is 14 the smallest order of a connected nontraceable locally hamiltonian graph? We answer the second question in the affirmative, but show that the correct number for the first question is 10. We develop a technique to construct connected locally hamiltonian and locally traceable graphs that are not traceable. We use this technique to construct such graphs with various prescribed properties.

Mots clés: locally traceable; locally hamiltonian; nonhamiltonian; nontraceable

Domaines: Mathématique discrète [cs.DM]

Johan de Wet, Susan van Aardt. Traceability of locally hamiltonian and locally traceable graphs. Discrete Mathematics and Theoretical Computer Science, 2016, Vol. 17 no. 3 (3), pp.245-262. ⟨10.46298/dmtcs.2144⟩. ⟨hal-01352838⟩