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The Chvátal-Erdős condition and 2-factors with a specified number of components

Guantao Chen, Ronald J. Gould, Ken-ichi Kawarabayashi, Katsuhiro Ota, Akira Saito, Ingo Schiermeyer (2007)

Discussiones Mathematicae Graph Theory

Let G be a 2-connected graph of order n satisfying α(G) = a ≤ κ(G), where α(G) and κ(G) are the independence number and the connectivity of G, respectively, and let r(m,n) denote the Ramsey number. The well-known Chvátal-Erdös Theorem states that G has a hamiltonian cycle. In this paper, we extend this theorem, and prove that G has a 2-factor with a specified number of components if n is sufficiently large. More precisely, we prove that (1) if n ≥ k·r(a+4, a+1), then G has a 2-factor with k components,...

The decomposability of additive hereditary properties of graphs

Izak Broere, Michael J. Dorfling (2000)

Discussiones Mathematicae Graph Theory

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. If ₁,...,ₙ are properties of graphs, then a (₁,...,ₙ)-decomposition of a graph G is a partition E₁,...,Eₙ of E(G) such that G [ E i ] , the subgraph of G induced by E i , is in i , for i = 1,...,n. We define ₁ ⊕...⊕ ₙ as the property G ∈ : G has a (₁,...,ₙ)-decomposition. A property is said to be decomposable if there exist non-trivial hereditary properties ₁ and ₂ such that = ₁⊕ ₂....

The edge domination problem

Shiow-Fen Hwang, Gerard J. Chang (1995)

Discussiones Mathematicae Graph Theory

An edge dominating set of a graph is a set D of edges such that every edge not in D is adjacent to at least one edge in D. In this paper we present a linear time algorithm for finding a minimum edge dominating set of a block graph.

The Fan-Raspaud conjecture: A randomized algorithmic approach and application to the pair assignment problem in cubic networks

Piotr Formanowicz, Krzysztof Tanaś (2012)

International Journal of Applied Mathematics and Computer Science

It was conjectured by Fan and Raspaud (1994) that every bridgeless cubic graph contains three perfect matchings such that every edge belongs to at most two of them. We show a randomized algorithmic way of finding Fan-Raspaud colorings of a given cubic graph and, analyzing the computer results, we try to find and describe the Fan-Raspaud colorings for some selected classes of cubic graphs. The presented algorithms can then be applied to the pair assignment problem in cubic computer networks. Another...

The list linear arboricity of planar graphs

Xinhui An, Baoyindureng Wu (2009)

Discussiones Mathematicae Graph Theory

The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. An and Wu introduce the notion of list linear arboricity lla(G) of a graph G and conjecture that lla(G) = la(G) for any graph G. We confirm that this conjecture is true for any planar graph having Δ ≥ 13, or for any planar graph with Δ ≥ 7 and without i-cycles for some i ∈ {3,4,5}. We also prove that ⌈½Δ(G)⌉ ≤ lla(G) ≤ ⌈½(Δ(G)+1)⌉ for any planar graph having Δ ≥ 9.

The maximum genus, matchings and the cycle space of a graph

Hung-Lin Fu, Martin Škoviera, Ming-Chun Tsai (1998)

Czechoslovak Mathematical Journal

In this paper we determine the maximum genus of a graph by using the matching number of the intersection graph of a basis of its cycle space. Our result is a common generalization of a theorem of Glukhov and a theorem of Nebeský .

The order of uniquely partitionable graphs

Izak Broere, Marietjie Frick, Peter Mihók (1997)

Discussiones Mathematicae Graph Theory

Let ₁,...,ₙ be properties of graphs. A (₁,...,ₙ)-partition of a graph G is a partition V₁,...,Vₙ of V(G) such that, for each i = 1,...,n, the subgraph of G induced by V i has property i . If a graph G has a unique (₁,...,ₙ)-partition we say it is uniquely (₁,...,ₙ)-partitionable. We establish best lower bounds for the order of uniquely (₁,...,ₙ)-partitionable graphs, for various choices of ₁,...,ₙ.

The Ryjáček Closure and a Forbidden Subgraph

Akira Saito, Liming Xiong (2016)

Discussiones Mathematicae Graph Theory

The Ryjáček closure is a powerful tool in the study of Hamiltonian properties of claw-free graphs. Because of its usefulness, we may hope to use it in the classes of graphs defined by another forbidden subgraph. In this note, we give a negative answer to this hope, and show that the claw is the only forbidden subgraph that produces non-trivial results on Hamiltonicity by the use of the Ryjáček closure.

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