Circuits through specified edges

Long cycles through specified edges and vertices

Spanning Cycles Through Specified Edges in Bipartite Graphs

Pósa proved that if G is an n-vertex graph in which any two nonadjacent vertices have degree sum at least n + k, then G has a spanning cycle containing any specified family of disjoint paths with a total of k edges. We consider the analogous problem for a bipartite graph G with n vertices and parts of equal size. Let F be a subgraph of G whose components are nontrivial paths. Let k be the numbe...

Title Number of Hamiltonian Circuits Containing Specified Edges in an Incom plete Graph and Its Application

A procedure for finding the number of Hamiltonian cir'cuits containing specined edges in a given incomplete graph is introduced and it is shown that the number of such Hamiltonian circuits can be expressed by a general fbrmula when the specified edges make a systematic form with all edges belonging to complement of the given incomplete graph. This procedure is applied to rnultigraphs with paral...

[a, b]-Factors Excluding Some Specified Edges In Graphs

Let G be a graph of order n, and let a, b and m be positive integers with 1 ≤ a < b. An [a, b]-factor of G is defined as a spanning subgraph F of G such that a ≤ dF (x) ≤ b for each x ∈ V (G). In this paper, it is proved that if n ≥ (a+b−1+ √ (a+b+1)m−2)2−1 b and δ(G) > n + a + b − 2√bn+ 1, then for any subgraph H of G with m edges, G has an [a, b]-factor F such that E(H) ∩ E(F ) = ∅. This resu...

Trees through specified vertices

We prove a conjecture of Horak that can be thought of as an extension of classical results including Dirac’s theorem on the existence of Hamiltonian cycles. Namely, we prove for 1 ≤ k ≤ n − 2 if G is a connected graph with A ⊂ V (G) such that dG(v) ≥ k for all v ∈ A, then there exists a subtree T of G such that V (T ) ⊃ A and dT (v) ≤ ⌈ n−1 k ⌉ for all v ∈ A.