We consider the number of vertex independent sets i(G). In general, the problem of determining the value of i(G) is NP -complete. We present several upper and lower bounds for i(G) in terms of order, size or independence number. We obtain improved bounds for i(G) on restricted graph classes such as the bipartite graphs, unicyclic graphs, regular graphs and claw-free graphs.

For every k ∈ N0, we consider graphs in which for any induced subgraph, ∆ ≤ χ − 1 + k holds, and call this family of graphs Υk, where ∆ is the maximum degree and χ is the chromatic number of the subgraph. We give a finite forbidden induced subgraph characterization for every k. The results are compared to those given in [6], where the graphs in which for any induced subgraph, ∆ ≤ ω − 1 + k hold...

We introduce a closure concept for 2-factors in claw-free graphs that generalizes the closure introduced by the first author. The 2-factor closure of a graph is uniquely determined and the closure operation turns a claw-free graph into the line graph of a graph containing no cycles of length at most 5 and no cycles of length 6 satisfying a certain condition. A graph has a 2-factor if and only i...

Recently, Jackson and Yoshimoto proved that every bridgeless simple graph Gwith δ(G) ≥ 3 has an even factor in which every component has order at least four, which strengthens a classical result of Petersen. In this paper, we give a strengthening of the above result and show that the above graphs have an even factor in which every component has order at least four that does not contain any give...

We consider the question of the range of the number of cycles possible in a 2-factor of a 2-connected claw-free graph with sufficiently high minimum degree. (By claw-free we mean the graph has no induced K1,3.) In particular, we show that for such a graph G of order n ≥ 51 with δ(G) ≥ n−2 3 , G contains a 2-factor with exactly k cycles, for 1 ≤ k ≤ n−24 3 . We also show that this result is shar...

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 Ryjáček, Saito, and Schelp (Closure, 2-factors, and cycle coverings in claw-free graphs, J. Graph Theory, 32 (1999), no. 2, 109-117) and show that the number ...

A graph G is a minimal claw-free graph (m.c.f. graph) if it contains no K1,3 (claw) as an induced subgraph and if, for each edge e of G, G − e contains an induced claw. We investigate properties of m.c.f. graphs, establish sharp bounds on their orders and the degrees of their vertices, and characterize graphs which have m.c.f. line graphs. MSC 2000: 05C75, 05C07

A graph is a quasi-line graph if for every vertex v, the set of neighbours of v is expressible as the union of two cliques. Such graphs are more general than line graphs, but less general than claw-free graphs. Here we give a construction for all quasi-line graphs; it turns out that there are basically two kinds of connected quasi-line graphs, one a generalization of line graphs, and the other ...

A fast algorithm to remove proper and homogenous pairs of cliques (while preserving some graph invariants) Abstract We introduce a family of reductions for removing proper and homogeneous pairs of cliques from a graph G. This family generalizes some routines presented in the literature, mostly in the context of claw-free graphs. These reductions can be embedded in a simple algorithm that in at ...