Graphs and Combinatorics (2018) 34:289–312
Minimal k-Connected Non-Hamiltonian Graphs
· Emily Marshall
Received: 24 April 2017 / Revised: 31 January 2018 / Published online: 14 February 2018
© Springer Japan KK, part of Springer Nature 2018
Abstract In this paper, we explore minimal k-connected non-Hamiltonian graphs.
Graphs are said to be minimal in the context of some containment relation; we focus
on subgraphs, induced subgraphs, minors, and induced minors. When k = 2, we
discuss all minimal 2-connected non-Hamiltonian graphs for each of these four rela-
tions. When k = 3, we conjecture a set of minimal non-Hamiltonian graphs for the
minor relation and we prove one case of this conjecture. In particular, we prove all
3-connected planar triangulations which do not contain the Herschel graph as a minor
Keywords Hamilton cycles · Graph minors
Hamilton cycles in graphs are cycles which visit every vertex of the graph. Determining
their existence in a graph is an NP-complete problem and as such, there is a large body
of research proving necessary and sufﬁcient conditions. In this paper, we analyze
non-Hamiltonian graphs. In particular, we consider the following general question:
The ﬁrst author was supported in part by NSF Grant DMS-1500699.
The work for this paper was largely done while the second author was at Louisiana State University.
Mathematics Department, Louisiana State University, Baton Rouge, LA 70803, USA
Computer Science and Mathematics Department, Arcadia University, Glenside, PA 19038, USA