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The flower conjecture in special classes of graphs

100%

Ryjáček Z. , Schiermeyer I.

Discussiones Mathematicae Graph Theory

1995

tom 15

nr 2

179-184

We say that a spanning eulerian subgraph F ⊂ G is a flower in a graph G if there is a vertex u ∈ V(G) (called the center of F) such that all vertices of G except u are of the degree exactly 2 in F. A graph G has the flower property if every vertex of G is a center of a flower. Kaneko conjectured that G has the flower property if and only if G is hamiltonian. In the present paper we prove this conjecture in several special classes of graphs, among others in squares and in a certain subclass of claw-free graphs.

Factors and circuits in $K_ {1,3}$-free graphs

100%

Ryjáček Z.

Banach Center Publications

1989

tom 25

nr 1

137-145

Distance-Locally Disconnected Graphs

80%

Miller M. , Ryan J. , Ryjáček Z.

Discussiones Mathematicae Graph Theory

2013

tom 33

nr 1

203-215

For an integer k ≥ 1, we say that a (finite simple undirected) graph G is k-distance-locally disconnected, or simply k-locally disconnected if, for any x ∈ V (G), the set of vertices at distance at least 1 and at most k from x induces in G a disconnected graph. In this paper we study the asymptotic behavior of the number of edges of a k-locally disconnected graph on n vertices. For general graphs, we show that this number is Θ(n2) for any fixed value of k and, in the special case of regular graphs, we show that this asymptotic rate of growth cannot be achieved. For regular graphs, we give a general upper bound and we show its asymptotic sharpness for some values of k. We also discuss some connections with cages.

On traceability and 2-factors in claw-free graphs

80%

Fronček D. , Ryjáček Z. , Skupień Z.

Discussiones Mathematicae Graph Theory

2004

tom 24

nr 1

55-71

If G is a claw-free graph of sufficiently large order n, satisfying a degree condition σₖ > n + k² - 4k + 7 (where k is an arbitrary constant), then G has a 2-factor with at most k - 1 components. As a second main result, we present classes of graphs 𝓒₁,...,𝓒₈ such that every sufficiently large connected claw-free graph satisfying degree condition σ₆(k) > n + 19 (or, as a corollary, δ(G) > (n+19)/6) either belongs to $⋃ ⁸_{i=1} 𝓒_i$ or is traceable.