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Associative graph products and their independence, domination and coloring numbers

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

Discussiones Mathematicae Graph Theory

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 of a graph...

Backbone colorings along stars and matchings in split graphs: their span is close to the chromatic number

Hajo Broersma, Bert Marchal, Daniel Paulusma, A.N.M. Salman (2009)

Discussiones Mathematicae Graph Theory

We continue the study on backbone colorings, a variation on classical vertex colorings that was introduced at WG2003. Given a graph G = (V,E) and a spanning subgraph H of G (the backbone of G), a λ-backbone coloring for G and H is a proper vertex coloring V→ {1,2,...} of G in which the colors assigned to adjacent vertices in H differ by at least λ. The algorithmic and combinatorial properties of backbone colorings have been studied for various types of backbones in a number of papers. The main outcome...

𝓟-bipartitions of minor hereditary properties

Piotr Borowiecki, Jaroslav Ivančo (1997)

Discussiones Mathematicae Graph Theory

We prove that for any two minor hereditary properties 𝓟₁ and 𝓟₂, such that 𝓟₂ covers 𝓟₁, and for any graph G ∈ 𝓟₂ there is a 𝓟₁-bipartition of G. Some remarks on minimal reducible bounds are also included.

Bounds for the b-Chromatic Number of Subgraphs and Edge-Deleted Subgraphs

P. Francis, S. Francis Raj (2016)

Discussiones Mathematicae Graph Theory

A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. In this paper, we obtain bounds for the b- chromatic number of induced subgraphs in terms of the b-chromatic number of the original graph. This turns out to be a generalization...

Bounds for the rainbow connection number of graphs

Ingo Schiermeyer (2011)

Discussiones Mathematicae Graph Theory

An edge-coloured graph G is rainbow-connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow-connected. In this paper we show some new bounds for the rainbow connection number of graphs depending on the minimum degree and other graph parameters. Moreover, we discuss sharpness of some of these bounds.

Broken Circuits in Matroids-Dohmen’s Inductive Proof

Wojciech Kordecki, Anna Łyczkowska-Hanćkowiak (2013)

Discussiones Mathematicae Graph Theory

Dohmen [4] gives a simple inductive proof of Whitney’s famous broken circuits theorem. We generalise his inductive proof to the case of matroids

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