Prescribed edges and forbidden edges for a cycle in a planar graph

On Hamiltonian Cycles through Prescribed Edges of a Planar Graph

We use [3] for terminology and notation not defined here and consider finite simple graphs only. The first major result on the existence of hamiltonian cycles in graphs embeddable in surfaces was by H. Whitney [12] in 1931, who proved that 4-connected maximal planar graphs are hamiltonian. In 1956, W.T. Tutte [10,11] generalized Whitney’s result from maximal planar graphs to arbitrary 4-connect...

Matching extensions with prescribed and forbidden edges

Suppose G connected graph on p vertices that contains perfect Then G is said to have property n) if p 2: 2(m + n + 1) and if for each pair of disjoint independent M, N E( G) of m, n there exists a perfect matching P in G such that M S;;;; P and 0. We discuss the circumstances under which E(m, n) =? E(x, y), and prove that (surprisingly) in general E(m, n) does not imply E(m, n-1).

The Maximum Number of Edges in a Strongly Multiplicative Graph

Inserting Multiple Edges into a Planar Graph

Let G be a connected planar (but not yet embedded) graph and F a set of additional edges not in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. An optimal solution to this problem is known to approximate the crossing number of the graph G+ F . Finding an exact solution to MEI is NP...

The Hamiltonian cycle and Hamiltonian paths passing through prescribed edges in a star graph with faulty edges

Let Fe be the set of faulty edges of Sn and E0 be the edge set of some pairwise vertex-disjoint paths of Sn. All edges of E0 lie on a Hamiltonian cycle of Sn−Fe, if |Fe| ≤ n−3, |E0| ≤ 2n−5−2|Fe| and lie on a Hamiltonian path P (u, v) where d(u, v) is odd, |Fe| ≤ n− 3, |E0| ≤ 2n− 7− 2|Fe| .