Hamilton-Connected Graph
A graph 
is Hamilton-connected if every two vertices of 
are connected by a Hamiltonian path (Bondy and Murty 1976, p. 61). In other words, a graph is Hamilton-connected if it has a 
Hamiltonian path for all pairs of vertices 
and 
. The illustration above shows a set of Hamiltonian paths that make the wheel graph 
hamilton-connected.
By definition, a graph with vertex count 
having a detour matrix whose off-diagonal elements are all equal to 
is Hamilton-connected. Conversely, any graph having a detour matrix with an off-diagonal element less than 
is not Hamilton-connected.
All Hamilton-connected graphs are Hamiltonian. All complete graphs are Hamilton-connected (with the trivial exception of the singleton graph), and all bipartite graphs are not Hamilton-connected.
Dupuis and Wagon (2014) conjectured that all non-bipartite Hamiltonian vertex-transitive graphs are Hamilton-connected except for odd cycle graphs 
with 
and the dodecahedral graph.
A simple algorithm for determining if a graph is Hamilton-connected proceeds as follows. For all pairs 
of vertices:
1. Add a new vertex 
.
2. Add new edges 
and 
.
3. If this graph is not Hamiltonian, return false; otherwise, continue to next pair.
If the algorithm checks all pairs without returning false, return true.
A small modification of a theorem due to Chvátal and Erdős establishes that if 
for a graph 
, where 
is the independence number and 
the vertex connectivity, then 
is Hamilton-connected (A. E. Brouwer, pers. comm., Dec. 17, 2012).
As a result of the theorem that for a connected regular graph 
on 
vertices with vertex degree 
and smallest graph eigenvalue 
,
it therefore follows that if
for a connected regular graph, the graph is Hamilton-connected (A. E. Brouwer, pers. comm., Dec. 17, 2012).
Every 8-connected claw-free graph is Hamilton-connected (Hu et al. 2005), as is every Johnson graph (Alspach 2013). Chen and Quimpo (1981) proved that a connected Cayley graph on a finite Abelian group of odd order with vertex degree at least three is Hamilton-connected.
Pensaert (2002) conjectured that for 
with 
, the generalized Petersen graph 
is Hamilton-laceable if 
is even and 
is odd, and Hamilton-connected otherwise.
The numbers of Hamilton-connected simple graphs on 
, 2, ... nodes are 1, 1, 1, 1, 3, 13, 116, ... (OEIS A057865), the first few of which are illustrated above.
Examples of Hamilton-connected graphs include antiprism graphs, complete graphs, Möbius ladders, prism graphs of odd order, wheel graphs, the truncated prism graph, truncated cubical graph, truncated tetrahedral graph, Grötzsch graph, Frucht graph, and Hoffman-Singleton graph.
REFERENCES
Alspach, B. "Johnson Graphs are Hamilton-Connected." Ars Math. Contemporanea 6, 21-23, 2013.
Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 61, 1976.
Chen, C. C. and Quimpo, N. F. "On Strongly Hamiltonian Abelian Group Graphs." In Combinatorial Mathematics. VIII. Proceedings of the Eighth Australian Conference held at Deakin University, Geelong, August 25-29, 1980
(Ed. K. L. McAvaney). Berlin: Springer-Verlag, pp. 23-34, 1981.
Dupuis, M. and Wagon, S. "Laceable Knights." To appear in Ars Math Contemp.Hu, Z.; Tian, F.; and Wei, B. "Hamilton Connectivity of Line Graphs and Claw-Free Graphs." J. Graph Th. 50, 130-141, 2005.
Pensaert, W. P. J. "Hamilton Paths in Generalized Petersen Graphs." Thesis. Waterloo, Ontario, Canada. January 2002.
http://etd.uwaterloo.ca/etd/wpjpensaert2002.pdf.Sloane, N. J. A. Sequence A057865 in "The On-Line Encyclopedia of Integer Sequences."
