On pseudo 2-factors

We show that a graph with minimum degree δ, independence number α≥δ and without isolated vertices, possesses a partition by vertex-disjoint cycles and at most α−δ+1 edges or vertices.     

On 2-factors with cycles containing specified edges in a bipartite graph

Let k≥1 be an integer and G=(V₁,V₂;E) a bipartite graph with |V₁|=|V₂|=n such that n≥2k+2. In this paper it has been proved that if for each pair of nonadjacent vertices x∈V₁ and y∈V₂, , then for any k independent edges e₁,…,e_k of G, G has a 2-factor with k+1 cycles C₁,…,C_k+1 such that e_i∈E(C_i) and |V(C_i)|=4 for each i∈{1,…,k}. We shall also show that the conditions in this paper are sharp.     

Non-separating 2-factors of an even-regular graph

For a 2-factor F of a connected graph G, we consider G−F, which is the graph obtained from G by removing all the edges of F. If G−F is connected, F is said to be a non-separating 2-factor. In this paper we study a sufficient condition for a 2r-regular connected graph G to have such a 2-factor. As a result, we show that a 2r-regular connected graph G has a non-separating 2-factor whenever the number of vertices of G does not exceed 2r²+r.     

Even subgraphs of bridgeless graphs and 2-factors of line graphs

By Petersen's theorem, a bridgeless cubic multigraph has a 2-factor. Fleischner generalised this result to bridgeless multigraphs of minimum degree at least three by showing that every such multigraph has a spanning even subgraph. Our main result is that every bridgeless simple graph with minimum degree at least three has a spanning even subgraph in which every component has at least four vertices. We deduce that if G is a simple bridgeless graph with n vertices and minimum degree at least three, then its line graph has a 2-factor with at most max{1,(3n-4)/10} components. This upper bound is best possible.     

Closure, 2-factors,and cycle coverings in claw-free graphs

In this article, we study cycle coverings and 2-factors of a claw-free graph and those of its closure, which has been defined by the first author (On a closure concept in claw-free graphs, J Combin Theory Ser B 70 (1997), 217–224). For a claw-free graph G and its closure cl(G), we prove: (1) V(G) is covered by k cycles in G if and only if V(cl(G)) is covered by k cycles of cl(G); and (2) G has a 2-factor with at most k components if and only if cl(G) has a 2-factor with at most k components. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 109–117, 1999     

2-边连通图的线图的P3因子

如果图G有一个生成子图使得这个生成子图的每一个分支都是3个点的路，则称G有P3-因子．本文证明了对任何一个2-边连通图G，只要G的边数能被3整除，则G的线图就有P3-因子。     

A sufficient condition for the existence of an anti-directed 2-factor in a directed graph

Let D be a directed graph with vertex set V, arc set A, and order n. The graph underlyingD is the graph obtained from D by replacing each arc (u,v)∈A by an undirected edge {u,v} and then replacing each double edge by a single edge. An anti-directed (hamiltonian) cycleH in D is a (hamiltonian) cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. An anti-directed 2-factor in D is a vertex-disjoint collection of anti-directed cycles in D that span V. It was proved in Busch et al. (submitted for publication) [3] that if the indegree and the outdegree of each vertex of D is greater than then D contains an anti-directed Hamilton cycle. In this paper we prove that given a directed graph D, the problem of determining whether D has an anti-directed 2-factor is NP-complete, and we use a proof technique similar to the one used in Busch et al. (submitted for publication) [3] to prove that if the indegree and the outdegree of each vertex of D is greater than then D contains an anti-directed 2-factor.     

Degree conditions for the partition of a graph into cycles, edges and isolated vertices

Let k,n be integers with 2≤k≤n, and let G be a graph of order n. We prove that if max{d_G(x),d_G(y)}≥(n−k+1)/2 for any x,y∈V(G) with x≠y and xy∉E(G), then G has k vertex-disjoint subgraphs H₁,…,H_k such that V(H₁)∪?∪V(H_k)=V(G) and H_i is a cycle or K₁ or K₂ for each 1≤i≤k, unless k=2 and G=C₅, or k=3 and G=K₁∪C₅.     

Sufficient conditions for the existence of a path‐factor which are related to odd components

In this article, we are concerned with sufficient conditions for the existence of a ‐factor. We prove that for , there exists such that if a graph G satisfies for all , then G has a ‐factor, where is the number of components C of with . On the other hand, we construct infinitely many graphs G having no ‐factor such that for all .     

Vertex‐disjoint cycles containing specified vertices in a bipartite graph

A minimum degree condition is given for a bipartite graph to contain a 2‐factor each component of which contains a previously specified vertex. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 145–166, 2004     

On degree sum conditions for 2-factors with a prescribed number of cycles

For a vertex subset $X$ of a graph $G$, let ${\Delta }_t\left(X\right)$ be the maximum value of the degree sums of the subsets of $X$ of size $t$. In this paper, we prove the following result: Let $k,m$ be positive integers, and let $G$ be an $m$-connected graph of order $n\ge 5k?2$. If ${\Delta }_2\left(X\right)\ge n$ for every independent set $X$ of size $?m∕k?+1$ in $G$, then $G$ has a 2-factor with exactly $k$ cycles. This is a common extension of the results obtained by Brandt et al. (1997) and Yamashita (2008), respectively.     

Planar graph colorings without short monochromatic cycles

It is well known that every planar graph G is 2‐colorable in such a way that no 3‐cycle of G is monochromatic. In this paper, we prove that G has a 2‐coloring such that no cycle of length 3 or 4 is monochromatic. The complete graph K₅ does not admit such a coloring. On the other hand, we extend the result to K₅‐minor‐free graphs. There are planar graphs with the property that each of their 2‐colorings has a monochromatic cycle of length 3, 4, or 5. In this sense, our result is best possible. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 25–38, 2004     

Pairs and triples of forbidden subgraphs and the existence of a 2-factor

Let be a set of connected graphs, each of which has order at least three, and suppose that there exist infinitely many connected -free graphs of minimum degree at least  two and all except for finitely many of them have a 2-factor. In [J. Graph Theory, 64 (2010), 250–266], we proved that if , then one of the members in is a star. In this article, we determine the remaining members of and hence give a complete characterization of the pairs and triples of forbidden subgraphs.     

Planar graphs without specific cycles are 2-degenerate

A graph G is k-degenerate if each subgraph of G has a vertex of degree at most k. It is known that every simple planar graph with girth 6, or equivalently without 3-, 4-, and 5-cycles, is 2-degenerate. In this work, we investigate for which k every planar graph without 4-, 6-, … , and 2k-cycles is 2-degenerate. We determine that k is 5 and the result is tight since the truncated dodecahedral graph is a 3-regular planar graph without 4-, 6-, and 8-cycles. As a related result, we also show that every planar graph without 4-, 6-, 9-, and 10-cycles is 2-degenerate.     

Explicit 2-Factorisations of the Odd Graph

In this note we show how 1-factors in the middle two layers of the discrete cube can be used to construct 2-factors in the Odd graph (the Kneser graph of (k − 1)-sets from a (2k − 1)-set). In particular, we use the lexical matchings of Kierstead and Trotter, and the modular matchings of Duffus, Kierstead and Snevily, to give explicit constructions of two different 2-factorisations of the Odd graph. This revised version was published online in September 2006 with corrections to the Cover Date.     

不含短圈的平面图的2- 距离染色

图的2- 距离染色是将图中距离不超过2 的点对染不同的色. 本文证明了g(G) > 5 且Δ(G) > 18的平面图G 有(Δ + 6)-2- 距离染色.     

A note on degree sum conditions for 2-factors with a prescribed number of cycles in bipartite graphs

Let $G$ be a balanced bipartite graph of order $2n\ge 4$, and let ${\sigma }_{1,1}\left(G\right)$ be the minimum degree sum of two non-adjacent vertices in different partite sets of $G$. In 1963, Moon and Moser proved that if ${\sigma }_{1,1}\left(G\right)\ge n+1$, then $G$ is hamiltonian. In this note, we show that if $k$ is a positive integer, then the Moon–Moser condition also implies the existence of a 2-factor with exactly $k$ cycles for sufficiently large graphs. In order to prove this, we also give a ${\sigma }_{1,1}$ condition for the existence of $k$ vertex-disjoint alternating cycles with respect to a chosen perfect matching in $G$.