By Petersen's theorem, a bridgeless cubic graph has a 2-factor. H. Fleischner extended this result to bridgeless graphs of minimum degree at least three by showing that every such graph has a spanning even subgraph. Our main result is that, under the stronger hypothesis of 3-edge-connectivity, we can find a spanning even subgraph in which every component has at least five vertices. We show that this is in some sense best possible by constructing an infinite family of This research was carried out while the second author was visiting Queen Mary, University of London.Journal of Graph Theory ᭧ 2009 Wiley Periodicals, Inc. 3738 JOURNAL OF GRAPH THEORY 3-edge-connected graphs in which every spanning even subgraph has a 5-cycle as a component.

Let X and Y be two disjoint sets of points in the plane such that |X| = |Y| and no three points ofXUF are on the same line. Then we can draw an alternating Hamilton cycle on X U Y in the plane which passes through alternately points of X and those of Y, whose edges are straight-line segments, and which contains at most |X| -1 crossings. Our proof gives an 0(n 2 logn) time algorithm for finding such an alternating Hamilton cycle, where n = |X|. Moreover we show that the above upper bound |X| -1 on crossing number is best possible for some configurations.

In this paper, we prove that if a claw-free graph G with minimum degree 4 has no maximal clique of two vertices, then G has a 2-factor with at most (|G| − 1)/4 components. This upper bound is best possible. Additionally, we give a family of claw-free graphs with minimum degree 4 in which every 2-factor contains more than n/ components.

(2009) 'Sharp upper bounds for the minimum number of components of 2-factors in claw-free graphs. ', Graphs and combinatorics., 25 (4). pp. 427-460. Further information on publisher's website:http://dx.doi.org/10.1007/s00373-009-0855-7Publisher's copyright statement:The nal publication is available at Springer via http://dx.doi.org/10.1007/s00373-009-0855-7Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. Let G be a claw-free graph with order n and minimum degree δ. We improve results of Faudree et al. and Gould & Jacobson, and solve two open problems by proving the following two results. If δ = 4, then G has a 2-factor with at most (5n − 14)/18 components, unless G belongs to a finite class of exceptional graphs. If δ ≥ 5, then G has a 2-factor with at most (n − 3)/(δ − 1) components, unless G is a complete graph. These bounds are best possible in the sense that we cannot replace 5/18 by a smaller quotient and we cannot replace δ − 1 by δ, respectively.

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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.

Let G be a simple graph with order n and minimum degree at least two. In this paper, we prove that if every odd branch-bond in G has an edge-branch, then its line graph has a 2-factor with at most 3n−2 8 components. For a simple graph with minimum degree at least three also, the same conclusion holds.

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