Noncrossing Hamiltonian paths in geometric graphs

Noncrossing Hamiltonian Paths in Geometric Graphs

Hamiltonian paths in Cayley graphs

The classical Lovász conjecture says that every connected Cayley graph is Hamiltonian. We present a short survey of various results in that direction and make some additional observations. In particular, we prove that every finite group G has a generating set of size at most log2 |G|, such that the corresponding Cayley graph contains a Hamiltonian cycle. We also present an explicit construction...

Hamiltonian paths on Platonic graphs

We develop a combinatorial method to show that the dodecahedron graph has, up to rotation and reflection, a unique Hamiltonian cycle. Platonic graphs with this property are called topologically uniquely Hamiltonian. The same method is used to demonstrate topologically distinct Hamiltonian cycles on the icosahedron graph and to show that a regular graph embeddable on the 2-holed torus is topolog...

Squeeze-free Hamiltonian Paths in Grid Graphs

Motivated by multi-robot construction systems, we introduce the problem of finding squeeze-free Hamiltonian paths in grid graphs. A Hamiltonian path is squeeze-free if it does not pass between two previously visited vertices lying on opposite sides. We determine necessary and sufficient conditions for the existence of squeeze-free Hamiltonian paths in staircase grid graphs. Our proofs are const...

Hamiltonian paths in L-shaped grid graphs

Grid graphs are subgraphs of the infinite 2-dimensional integer grid. The Hamiltonian path problem for general grid graphs is a well-known NP-complete problem. In this paper, we present necessary and sufficient conditions for the existence of a Hamiltonian path between two given vertices in L−shaped grid graphs. We also show that a Hamiltonian path between two given vertices of a L−shaped grid ...