On cycles through specified vertices

Pairs of forbidden class of subgraphs concerning K1, 3 and P6 to have a cycle containing specified vertices

In [3], Faudree and Gould showed that if a 2-connected graph contains no K1,3 and P6 as an induced subgraph, then the graph is hamiltonian. In this paper, we consider the extension of this result to cycles passing through specified vertices. We define the families of graphs which are extension of the forbidden pair K1,3 and P6, and prove that the forbidden families implies the existence of cycl...

Forbidden subgraphs and the existence of paths and cycles passing through specified vertices

In [2], Duffus et al. showed that every connected graph G which contains no induced subgraph isomorphic to a claw or a net is traceable. And they also showed that if a 2-connected graph G satisfies the above conditions, then G is hamiltonian. In this paper, modifying the conditions of Duffus et al.’s theorems, we give forbidden structures for a specified set of vertices which assures the existe...

Long cycles through specified edges and vertices

Chorded Cycles

A chord is an edge between two vertices of a cycle that is not an edge on the cycle. If a cycle has at least one chord, then the cycle is called a chorded cycle, and if a cycle has at least two chords, then the cycle is called a doubly chorded cycle. The minimum degree and the minimum degree-sum conditions are given for a graph to contain vertex-disjoint chorded (doubly chorded) cycles containi...

Cycles through specified vertices in triangle-free graphs

Let G be a triangle-free graph with δ(G) ≥ 2 and σ4(G) ≥ |V (G)|+ 2. Let S ⊂ V (G) consist of less than σ4/4 + 1 vertices. We prove the following. If all vertices of S have degree at least three, then there exists a cycle C containing S. Both the upper bound on |S| and the lower bound on σ4 are best possible.