The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer k ≥ 0, to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback verte...

The well-known Erdős-Pósa theorem says that for any integer k and any graph G, either G contains k vertexdisjoint cycles or a vertex setX of order at most c·k log k (for some constant c) such that G−X is a forest. Thomassen [39] extended this result to the even cycles, but on the other hand, it is well-known that this theorem is no longer true for the odd cycles. However, Reed [31] proved that ...

In this paper we provide some sufficient conditions for the existence of an odd or even cycle that passing a given vertex or an edge in 2-connected or 2-edge connected graphs. We provide some similar conditions for the existence of an odd or even circuit that passing a given vertex or an edge in 2-edge connected graphs. We show that if G is a 2-connected k-regular graph, k ≥ 3, then every edge ...

Let X be a subset of the vertex set of a graph G. We denote by (X) the smallest number of vertices separating two vertices of X if X does not induce a complete subgraph of G, otherwise we put (X) = |X| − 1 if |X| 2 and (X) = 1 if |X| = 1. We prove that if (X) 2 then every set of at most (X) vertices of X is contained in a cycle of G. Thus, we generalize a similar result of Dirac. Applying this ...

Let G be a 2-connected graph of order n. For any u ∈ V (G) and l ∈ {m,m + 1, . . . , n}, if G has a cycle of length l, then G is called [m,n]pancyclic, and if G has a cycle of length l which contains u, then G is called [m,n]-vertex pancyclic. Let δ(G) be a minimum degree ofG and let NG(x) be the neighborhood of a vertex x in G. In [Australas. J. Combin. 12 (1995), 81–91] Liu, Lou and Zhao prov...

A special case of a conjecture of M. El-Zahár states that a graph G with 2k vertices and minimum degree k, contains every bipartite 2-regular graph H on 2k vertices as a spanning subgraph. In this paper it will be proved that G contains the union of (k− 1) cycles of lengths 4 and a path of order 4. Using this result it will also be proved that G contains the union of (k−2) cycles of lengths 4 a...

We consider a simple, undirected graph G. The ball of a subset Y of vertices in G is the set of vertices in G at distance at most one from a vertex in Y . Assuming that the balls of all subsets of at most two vertices in G are distinct, we prove that G admits a cycle with length at least 7.

A diagonal cycle is a cycle of length 2n together with one edge between two vertices at distance n in the cycle. For example, /{4 e is a cycle of length fonr with one extra edge between two vertices at distance two in the cycle. Methods for decomposing the complete graph Kv into diagonal cycles are given when v == 0 (mod 2n + 1) if n is odel, and when v == 1 (mod 2n + 1), if n 3 (mod 4); in the...

The vertex arboricity $rho(G)$ of a graph $G$ is the minimum number of subsets into which the vertex set $V(G)$ can be partitioned so that each subset induces an acyclic graph‎. ‎A graph $G$ is called list vertex $k$-arborable if for any set $L(v)$ of cardinality at least $k$ at each vertex $v$ of $G$‎, ‎one can choose a color for each $v$ from its list $L(v)$ so that the subgraph induced by ev...

Corrádi and Hajnal [1] showed that any graph of order at least 3k with minimum degree at least 2k contains k vertex-disjoint cycles. This minimum degree condition is sharp, because the complete bipartite graph K2k−1,n−2k+1 does not contain k vertex-disjoint cycles. About the existence of vertex-disjoint cycles of the same length, Thomassen [4] conjectured that the same minimum degree condition ...