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2-factors in claw-free graphs

100%

Chen G., Faudree J., Gould R., Saito A.

Discussiones Mathematicae Graph Theory

2000

tom 20

nr 2

165-172

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 $K_{1,3}$.) 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 sharp in the sense that if we lower δ(G), we cannot obtain the full range of values for k.

Chvátal-Erdös type theorems

88%

Faudree J., Faudree R., Gould R., Jacobson M., Magnant C.

Discussiones Mathematicae Graph Theory

2010

tom 30

nr 2

245-256

The Chvátal-Erdös theorems imply that if G is a graph of order n ≥ 3 with κ(G) ≥ α(G), then G is hamiltonian, and if κ(G) > α(G), then G is hamiltonian-connected. We generalize these results by replacing the connectivity and independence number conditions with a weaker minimum degree and independence number condition in the presence of sufficient connectivity. More specifically, it is noted that if G is a graph of order n and k ≥ 2 is a positive integer such that κ(G) ≥ k, δ(G) > (n+k²-k)/(k+1), and δ(G) ≥ α(G)+k-2, then G is hamiltonian. It is shown that if G is a graph of order n and k ≥ 3 is a positive integer such that κ(G) ≥ 4k²+1, δ(G) > (n+k²-2k)/k, and δ(G) ≥ α(G)+k-2, then G is hamiltonian-connected. This result supports the conjecture that if G is a graph of order n and k ≥ 3 is a positive integer such that κ(G) ≥ k, δ(G) > (n+k²-2k)/k, and δ(G) ≥ α(G)+k-2, then G is hamiltonian-connected, and the conjecture is verified for k = 3 and 4.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 Faudree J.

2 Gould R.

1 Chen G.

1 Faudree R.

1 Jacobson M.

1 Magnant C.

1 Saito A.

1 2010

1 2000

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2-factors in claw-free graphs

Chvátal-Erdös type theorems