All a b c d e f g h i j k l m n o p q r s t u v w x y z Other

Previous Page 25 Next

Displaying 481 – 500 of 719

Showing per page

On the cost chromatic number of outerplanar, planar, and line graphs

John Mitchem, Patrick Morriss, Edward Schmeichel (1997)

Discussiones Mathematicae Graph Theory

We consider vertex colorings of graphs in which each color has an associated cost which is incurred each time the color is assigned to a vertex. The cost of the coloring is the sum of the costs incurred at each vertex. The cost chromatic number of a graph with respect to a cost set is the minimum number of colors necessary to produce a minimum cost coloring of the graph. We show that the cost chromatic number of maximal outerplanar and maximal planar graphs can be arbitrarily large and construct...

On the dominator colorings in trees

Houcine Boumediene Merouane, Mustapha Chellali (2012)

Discussiones Mathematicae Graph Theory

In a graph G, a vertex is said to dominate itself and all its neighbors. A dominating set of a graph G is a subset of vertices that dominates every vertex of G. The domination number γ(G) is the minimum cardinality of a dominating set of G. A proper coloring of a graph G is a function from the set of vertices of the graph to a set of colors such that any two adjacent vertices have different colors. A dominator coloring of a graph G is a proper coloring such that every vertex of V dominates all vertices...

On the factorization of reducible properties of graphs into irreducible factors

P. Mihók, R. Vasky (1995)

Discussiones Mathematicae Graph Theory

A hereditary property R of graphs is said to be reducible if there exist hereditary properties P₁,P₂ such that G ∈ R if and only if the set of vertices of G can be partitioned into V(G) = V₁∪V₂ so that ⟨V₁⟩ ∈ P₁ and ⟨V₂⟩ ∈ P₂. The problem of the factorization of reducible properties into irreducible factors is investigated.

On the heterochromatic number of circulant digraphs

Hortensia Galeana-Sánchez, Víctor Neumann-Lara (2004)

Discussiones Mathematicae Graph Theory

The heterochromatic number hc(D) of a digraph D, is the minimum integer k such that for every partition of V(D) into k classes, there is a cyclic triangle whose three vertices belong to different classes. For any two integers s and n with 1 ≤ s ≤ n, let D n , s be the oriented graph such that V ( D n , s ) is the set of integers mod 2n+1 and A ( D n , s ) = ( i , j ) : j - i ∈ 1 , 2 , . . . , n ∖ s . . In this paper we prove that h c ( D n , s ) ≤ 5 for n ≥ 7. The bound is tight since equality holds when s ∈ n,[(2n+1)/3].

On the Independence Number of Edge Chromatic Critical Graphs

Shiyou Pang, Lianying Miao, Wenyao Song, Zhengke Miao (2014)

Discussiones Mathematicae Graph Theory

In 1968, Vizing conjectured that for any edge chromatic critical graph G = (V,E) with maximum degree △ and independence number α (G), α (G) ≤ [...] . It is known that α (G) < [...] |V |. In this paper we improve this bound when △≥ 4. Our precise result depends on the number n2 of 2-vertices in G, but in particular we prove that α (G) ≤ [...] |V | when △ ≥ 5 and n2 ≤ 2(△− 1)

On the number of Russell’s socks or 2 + 2 + 2 + ⋯ = ?

Horst Herrlich, Eleftherios Tachtsis (2006)

Commentationes Mathematicae Universitatis Carolinae

The following question is analyzed under the assumption that the Axiom of Choice fails badly: Given a countable number of pairs of socks, then how many socks are there? Surprisingly this number is not uniquely determined by the above information, thus giving rise to the concept of Russell-cardinals. It will be shown that: • some Russell-cardinals are even, but others fail to be so; • no Russell-cardinal is odd; • no Russell-cardinal is comparable with any cardinal of the form ℵ α or 2 ℵ α ; • finite sums...

On the rainbow connection of Cartesian products and their subgraphs

Sandi Klavžar, Gašper Mekiš (2012)

Discussiones Mathematicae Graph Theory

Rainbow connection number of Cartesian products and their subgraphs are considered. Previously known bounds are compared and non-existence of such bounds for subgraphs of products are discussed. It is shown that the rainbow connection number of an isometric subgraph of a hypercube is bounded above by the rainbow connection number of the hypercube. Isometric subgraphs of hypercubes with the rainbow connection number as small as possible compared to the rainbow connection of the hypercube are constructed....

On the Rainbow Vertex-Connection

Xueliang Li, Yongtang Shi (2013)

Discussiones Mathematicae Graph Theory

A vertex-colored graph is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertexconnected. It was proved that if G is a graph of order n with minimum degree δ, then rvc(G) < 11n/δ. In this paper, we show that rvc(G) ≤ 3n/(δ+1)+5 for [xxx] and n ≥ 290, while rvc(G) ≤ 4n/(δ + 1) + 5...

Currently displaying 481 – 500 of 719