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A graph X is 2-spanning cyclable if for any pair of distinct vertices u and v there is a 2-factor of X consisting of two cycles such that u and v belong to distinct cycles. In this paper we examine the 2-spanning cyclability of honeycomb toroidal ...

For a graph H, let δ ( H ) be the minimum degree and let c ( H ) be the length of a longest cycle in H. A 2-factor of H is a spanning subgraph of H in which every component is a cycle. In [Discrete Math. 313 (2013) 1934-1943], Čada and Chiba ...

For a given graph R, a graph G is R-free if G does not contain R as an induced subgraph. It is known that every 2-tough graph with at least three vertices has a 2-factor. In graphs with restricted structures, it was shown that every 2 K 2-free 3/...

For a vertex u in a graph and a given positive integer k, let M k ( u ) denote the set of vertices whose distance from u is at most k. A graph satisfies the local Dirac's condition if the degree of each vertex u in it is at least | M 2 ...

A graph G admitting a 2-factor is pseudo 2-factor isomorphic if the parity of the number of cycles in all its 2-factors is the same. In Abreu et al. (2008) some of the authors of this note gave a partial characterisation of pseudo 2-factor ...

We investigate 2-planarizing 2-factors, i.e. 2-factors of embedded graphs so that cutting along the cycles of the 2-factor we get two plane graphs where the cycles of the 2-factors are a spanning set of face boundaries in each of the graphs. We ...

A graph is called H-free if it has no induced subgraph isomorphic to H. A graph is called $$N^i$$Ni-locally connected if $$G[\{ x\in V(G): 1\le d_G(w, x)\le i\}]$$G[{x?V(G):1≤dG(w,x)≤i}] is connected and $$N_2$$N2-locally connected if $$G[\{uv: \{uw, vw\...

A spanning subgraph F of a graph G is called an even factor of G if each vertex of F has even degree at least 2 in F. It was conjectured that if a graph G has an even factor, then it has an even factor F with $$|E(F)|\ge {4\over 7}(|E(G)| + 1)+ {2\over ...

We give a sufficient degree condition for the hamiltonicity of graphs with a 2-factor which is best possible in the same sense that Chvátal's well-known hamiltonian degree condition is best possible.

It is known that a 3-edge-connected graph has a spanning even subgraph in which every component contains at least five vertices, and the lower bound is best possible. A natural question arises of whether we can improve the lower bound by changing the ...

The paper considers the minimization of the sum of weights of edges forming a subset Uź ź U of the set of disjoint simple cycles at vertices ź ∈ V in graph H = (V,U) and covering V. The problem under study (2-f problem) is polynomially solvable in ...

A graph G is pseudo 2-factor isomorphic if the parity of the number of cycles in a 2-factor is the same for all 2-factors of G. Abreu etal. conjectured that K3,3, the Heawood graph and the Pappus graph are the only essentially 4-edge-connected pseudo 2-...

We study the complexity of finding 2-factors with various restrictions as well as edge-decompositions in (the underlying graphs of) digraphs. In particular we show that it is NP-complete to decide whether the underlying undirected graph of a digraph D ...

Let G = G n ( a 1 , a 2 ) be a connected circulant digraph of order n with two distinct jumps a 1 , a 2 < n . We give several sufficient conditions for a decomposition of G n ( a 1 , a 2 ) into directed cycles of equal lengths. We then prove that G n ( ...

We show that every 2-edge-connected cubic graph $G$ not isomorphic to the Petersen graph has a 2-factor with at most 2(n-2)/15 circuits of length 5, where $n$ is the number of vertices of $G$. We construct an infinite family of graphs, for which this ...

For a graph G , we denote by ( G ) the minimum degree of G . A graph G is said to be claw-free if G has no induced subgraph isomorphic to K _{1, 3}. In this article, we prove that every claw-free graph G with minimum degree at least 4 has a 2-...

A 2-factor in a graph is a spanning 2-regular subgraph, or equivalently a spanning collection of disjoint cycles. In this paper we investigate the existence of 2-factors with a bounded number of odd cycles in a graph. We extend results of Ryjacek, Saito,...

In this paper, we consider conditions that ensure a hamiltonian graph has a 2-factor with exactly k cycles. Brandt et al. proved that if G is a graph on n>=4k vertices with minimum degree at least n2, then G contains a 2-factor with exactly k cycles; ...

Let $G$ be an edge-colored graph. The minimum color degree $\delta^c(G)$ of $G$ is the largest integer $k$ such that for every vertex $v$, there are at least $k$ distinct colors on edges incident to $v$. We say that $G$ is properly colored if no two ...

An antimagic labeling of a finite simple undirected graph with q edges is a bijection from the set of edges to the set of integers {1,2,...,q} such that the vertex sums are pairwise distinct, where the vertex sum at vertex u is the sum of labels of all ...