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Chvátal's Condition cannot hold for both a graph and its complement

100%

Kostochka A., West D.

Discussiones Mathematicae Graph Theory

2006

tom 26

nr 1

73-76

Chvátal's Condition is a sufficient condition for a spanning cycle in an n-vertex graph. The condition is that when the vertex degrees are d₁, ...,dₙ in nondecreasing order, i < n/2 implies that $d_i > i$ or $d_{n-i} ≥ n-i$. We prove that this condition cannot hold in both a graph and its complement, and we raise the problem of finding its asymptotic probability in the random graph with edge probability 1/2.

Decompositions of quadrangle-free planar graphs

81%

Borodin O., Ivanova A., Kostochka A., Sheikh N.

Discussiones Mathematicae Graph Theory

2009

tom 29

nr 1

87-99

W. He et al. showed that a planar graph not containing 4-cycles can be decomposed into a forest and a graph with maximum degree at most 7. This degree restriction was improved to 6 by Borodin et al. We further lower this bound to 5 and show that it cannot be improved to 3.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 Kostochka A.

1 Borodin O.

1 Ivanova A.

1 Sheikh N.

1 West D.

1 2009

1 2006

Wyniki wyszukiwania

Chvátal's Condition cannot hold for both a graph and its complement

Decompositions of quadrangle-free planar graphs