On pseudo 2-factors

AbstractWe 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 Maximum Cycle Packings in Polyhedral Graphs

This paper addresses upper and lower bounds for the cardinality of a maximum vertex-/edge-disjoint cycle packing in a polyhedral graph G. Bounds on the cardinality of such packings are provided, that depend on the size, the order or the number of faces of G, respectively. Polyhedral graphs are constructed, that attain these bounds

Graphs with many vertex-disjoint cycles

We study graphs G in which the maximum number of vertex-disjoint cycles \nu(G) is close to the cyclomatic number \mu(G) which is a natural upper bound for \nu(G). Our main result is the existence of a finite set P(k) of graphs for all k \in \mathbb{N}_0 such that every 2-connected graph G with \mu(G) \nu (G) = k arises by applying a simple extension rule to a graph in \mathcal{P}(k). As an algorithmic consequence we describe algorithms calculating min{\mu(G)-\nu(G); k + 1} in linear time for fixed k

Packing disjoint cycles over vertex cuts

AbstractFor a graph G, let ν(G) and ν′(G) denote the maximum cardinalities of packings of vertex-disjoint and edge-disjoint cycles of G, respectively. We study the interplay of these two parameters and vertex cuts in graphs. If G is a graph whose vertex set can be partitioned into three non-empty sets S, V1, and V2 such that there is no edge between V1 and V2, and k=|S|, then our results imply that ν(G) is uniquely determined by the values ν(H) for at most 2k+1k!2 graphs H of order at most max{|V1|,|V2|}+k, and ν′(G) is uniquely determined by the values ν′(H) for at most 2k2+1 graphs H of order at most max{|V1|,|V2|}+k

Independence Number and Disjoint Theta Graphs

The goal of this paper is to find vertex disjoint even cycles in graphs. For this purpose, define a θ-graph to be a pair of vertices u,v with three internally disjoint paths joining u to v. Given an independence number α and a fixed integer k, the results contained in this paper provide sharp bounds on the order f(k,α) of a graph with independence number α(G)≤α which contains no k disjoint θ-graphs. Since every θ-graph contains an even cycle, these results provide k disjoint even cycles in graphs of order at least f(k,α)+1. We also discuss the relationship between this problem and a generalized ramsey problem involving sets of graphs