Displaying similar documents to “Weak k-reconstruction of Cartesian products”

Associative graph products and their independence, domination and coloring numbers

Richard J. Nowakowski, Douglas F. Rall (1996)

Discussiones Mathematicae Graph Theory

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Associative products are defined using a scheme of Imrich & Izbicki [18]. These include the Cartesian, categorical, strong and lexicographic products, as well as others. We examine which product ⊗ and parameter p pairs are multiplicative, that is, p(G⊗H) ≥ p(G)p(H) for all graphs G and H or p(G⊗H) ≤ p(G)p(H) for all graphs G and H. The parameters are related to independence, domination and irredundance. This includes Vizing's conjecture directly, and indirectly the Shannon capacity...

On Vizing's conjecture

Bostjan Bresar (2001)

Discussiones Mathematicae Graph Theory

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A dominating set D for a graph G is a subset of V(G) such that any vertex in V(G)-D has a neighbor in D, and a domination number γ(G) is the size of a minimum dominating set for G. For the Cartesian product G ⃞ H Vizing's conjecture [10] states that γ(G ⃞ H) ≥ γ(G)γ(H) for every pair of graphs G,H. In this paper we introduce a new concept which extends the ordinary domination of graphs, and prove that the conjecture holds when γ(G) = γ(H) = 3.

Highly connected counterexamples to a conjecture on α-domination

Zsolt Tuza (2005)

Discussiones Mathematicae Graph Theory

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An infinite class of counterexamples is given to a conjecture of Dahme et al. [1] concerning the minimum size of a dominating vertex set that contains at least a prescribed proportion of the neighbors of each vertex not belonging to the set.

Vertex-disjoint copies of K¯₄

Ken-ichi Kawarabayashi (2004)

Discussiones Mathematicae Graph Theory

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Let G be a graph of order n. Let K¯ₗ be the graph obtained from Kₗ by removing one edge. In this paper, we propose the following conjecture: Let G be a graph of order n ≥ lk with δ(G) ≥ (n-k+1)(l-3)/(l-2)+k-1. Then G has k vertex-disjoint K¯ₗ. This conjecture is motivated by Hajnal and Szemerédi's [6] famous theorem. In this paper, we verify this conjecture for l=4.

The crossing numbers of certain Cartesian products

Marián Klešč (1995)

Discussiones Mathematicae Graph Theory

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In this article we determine the crossing numbers of the Cartesian products of given three graphs on five vertices with paths.

The Domination Number of K 3 n

John Georges, Jianwei Lin, David Mauro (2014)

Discussiones Mathematicae Graph Theory

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Let K3n denote the Cartesian product Kn□Kn□Kn, where Kn is the complete graph on n vertices. We show that the domination number of K3n is [...]

Detour chromatic numbers

Marietjie Frick, Frank Bullock (2001)

Discussiones Mathematicae Graph Theory

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The nth detour chromatic number, χₙ(G) of a graph G is the minimum number of colours required to colour the vertices of G such that no path with more than n vertices is monocoloured. The number of vertices in a longest path of G is denoted by τ( G). We conjecture that χₙ(G) ≤ ⎡(τ(G))/n⎤ for every graph G and every n ≥ 1 and we prove results that support the conjecture. We also present some sufficient conditions for a graph to have nth chromatic number at most 2.

An Oriented Version of the 1-2-3 Conjecture

Olivier Baudon, Julien Bensmail, Éric Sopena (2015)

Discussiones Mathematicae Graph Theory

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The well-known 1-2-3 Conjecture addressed by Karoński, Luczak and Thomason asks whether the edges of every undirected graph G with no isolated edge can be assigned weights from {1, 2, 3} so that the sum of incident weights at each vertex yields a proper vertex-colouring of G. In this work, we consider a similar problem for oriented graphs. We show that the arcs of every oriented graph −G⃗ can be assigned weights from {1, 2, 3} so that every two adjacent vertices of −G⃗ receive distinct...

Products Of Digraphs And Their Competition Graphs

Martin Sonntag, Hanns-Martin Teichert (2016)

Discussiones Mathematicae Graph Theory

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If D = (V, A) is a digraph, its competition graph (with loops) CGl(D) has the vertex set V and {u, v} ⊆ V is an edge of CGl(D) if and only if there is a vertex w ∈ V such that (u, w), (v, w) ∈ A. In CGl(D), loops {v} are allowed only if v is the only predecessor of a certain vertex w ∈ V. For several products D1 ⚬ D2 of digraphs D1 and D2, we investigate the relations between the competition graphs of the factors D1, D2 and the competition graph of their product D1 ⚬ D2.

Vizing's conjecture and the one-half argument

Bert Hartnell, Douglas F. Rall (1995)

Discussiones Mathematicae Graph Theory

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The domination number of a graph G is the smallest order, γ(G), of a dominating set for G. A conjecture of V. G. Vizing [5] states that for every pair of graphs G and H, γ(G☐H) ≥ γ(G)γ(H), where G☐H denotes the Cartesian product of G and H. We show that if the vertex set of G can be partitioned in a certain way then the above inequality holds for every graph H. The class of graphs G which have this type of partitioning includes those whose 2-packing number is no smaller than γ(G)-1 as...