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M 2 -Edge Colorings Of Cacti And Graph Joins

Július Czap, Peter Šugerek, Jaroslav Ivančo (2016)

Discussiones Mathematicae Graph Theory

An edge coloring φ of a graph G is called an M2-edge coloring if |φ(v)| ≤ 2 for every vertex v of G, where φ(v) is the set of colors of edges incident with v. Let 𝒦2(G) denote the maximum number of colors used in an M2-edge coloring of G. In this paper we determine 𝒦2(G) for trees, cacti, complete multipartite graphs and graph joins.

Majority choosability of 1-planar digraph

Weihao Xia, Jihui Wang, Jiansheng Cai (2023)

Czechoslovak Mathematical Journal

A majority coloring of a digraph D with k colors is an assignment π : V ( D ) { 1 , 2 , , k } such that for every v V ( D ) we have π ( w ) = π ( v ) for at most half of all out-neighbors w N + ( v ) . A digraph D is majority k -choosable if for any assignment of lists of colors of size k to the vertices, there is a majority coloring of D from these lists. We prove that if U ( D ) is a 1-planar graph without a 4-cycle, then D is majority 3-choosable. And we also prove that every NIC-planar digraph is majority 3-choosable.

Markov convexity and local rigidity of distorted metrics

Manor Mendel, Assaf Naor (2013)

Journal of the European Mathematical Society

It is shown that a Banach space admits an equivalent norm whose modulus of uniform convexity has power-type p if and only if it is Markov p -convex. Counterexamples are constructed to natural questions related to isomorphic uniform convexity of metric spaces, showing in particular that tree metrics fail to have the dichotomy property.

Maximal graphs with respect to hereditary properties

Izak Broere, Marietjie Frick, Gabriel Semanišin (1997)

Discussiones Mathematicae Graph Theory

A property of graphs is a non-empty set of graphs. A property P is called hereditary if every subgraph of any graph with property P also has property P. Let P₁, ...,Pₙ be properties of graphs. We say that a graph G has property P₁∘...∘Pₙ if the vertex set of G can be partitioned into n sets V₁, ...,Vₙ such that the subgraph of G induced by Vi has property P i ; i = 1,..., n. A hereditary property R is said to be reducible if there exist two hereditary properties P₁ and P₂ such that R = P₁∘P₂. If P...

Maximal hypergraphs with respect to the bounded cost hereditary property

Ewa Drgas-Burchardt, Anna Fiedorowicz (2005)

Discussiones Mathematicae Graph Theory

The hereditary property of hypergraphs generated by the cost colouring notion is considered in the paper. First, we characterize all maximal graphs with respect to this property. Second, we give the generating function for the sequence describing the number of such graphs with the numbered order. Finally, we construct a maximal hypergraph for each admissible number of vertices showing some density property. All results can be applied to the problem of information storage.

Maximum Edge-Colorings Of Graphs

Stanislav Jendrol’, Michaela Vrbjarová (2016)

Discussiones Mathematicae Graph Theory

An r-maximum k-edge-coloring of G is a k-edge-coloring of G having a property that for every vertex v of degree dG(v) = d, d ≥ r, the maximum color, that is present at vertex v, occurs at v exactly r times. The r-maximum index [...] χr′(G) χ r ' ( G ) is defined to be the minimum number k of colors needed for an r-maximum k-edge-coloring of graph G. In this paper we show that [...] χr′(G)≤3 χ r ' ( G ) 3 for any nontrivial connected graph G and r = 1 or 2. The bound 3 is tight. All graphs G with [...] χ1′(G)=i χ 1 ' ( G ) = i , i...

Measure-theoretic unfriendly colorings

Clinton T. Conley (2014)

Fundamenta Mathematicae

We consider the problem of finding a measurable unfriendly partition of the vertex set of a locally finite Borel graph on standard probability space. After isolating a sufficient condition for the existence of such a partition, we show how it settles the dynamical analog of the problem (up to weak equivalence) for graphs induced by free, measure-preserving actions of groups with designated finite generating set. As a corollary, we obtain the existence of translation-invariant random unfriendly colorings...

Metric spaces with point character equal to their size

C. Avart, P. Komjath, Vojtěch Rödl (2010)

Commentationes Mathematicae Universitatis Carolinae

In this paper we consider the point character of metric spaces. This parameter which is a uniform version of dimension, was introduced in the context of uniform spaces in the late seventies by Jan Pelant, Cardinal reflections and point-character of uniformities, Seminar Uniform Spaces (Prague, 1973–1974), Math. Inst. Czech. Acad. Sci., Prague, 1975, pp. 149–158. Here we prove for each cardinal κ , the existence of a metric space of cardinality and point character κ . Since the point character can...

Minimal rankings of the Cartesian product Kₙ ☐ Kₘ

Gilbert Eyabi, Jobby Jacob, Renu C. Laskar, Darren A. Narayan, Dan Pillone (2012)

Discussiones Mathematicae Graph Theory

For a graph G = (V, E), a function f:V(G) → 1,2, ...,k is a k-ranking if f(u) = f(v) implies that every u - v path contains a vertex w such that f(w) > f(u). A k-ranking is minimal if decreasing any label violates the definition of ranking. The arank number, ψ r ( G ) , of G is the maximum value of k such that G has a minimal k-ranking. We completely determine the arank number of the Cartesian product Kₙ ☐ Kₙ, and we investigate the arank number of Kₙ ☐ Kₘ where n > m.

Minimal reducible bounds for hom-properties of graphs

Amelie Berger, Izak Broere (1999)

Discussiones Mathematicae Graph Theory

Let H be a fixed finite graph and let → H be a hom-property, i.e. the set of all graphs admitting a homomorphism into H. We extend the definition of → H to include certain infinite graphs H and then describe the minimal reducible bounds for → H in the lattice of additive hereditary properties and in the lattice of hereditary properties.

Minimum vertex ranking spanning tree problem for chordal and proper interval graphs

Dariusz Dereniowski (2009)

Discussiones Mathematicae Graph Theory

A vertex k-ranking of a simple graph is a coloring of its vertices with k colors in such a way that each path connecting two vertices of the same color contains a vertex with a bigger color. Consider the minimum vertex ranking spanning tree (MVRST) problem where the goal is to find a spanning tree of a given graph G which has a vertex ranking using the minimal number of colors over vertex rankings of all spanning trees of G. K. Miyata et al. proved in [NP-hardness proof and an approximation algorithm...

Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs

Hortensia Galeana-Sánchez, R. Rojas-Monroy, B. Zavala (2009)

Discussiones Mathematicae Graph Theory

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices of N there is no monochromatic path between them and for every vertex v ∉ N there is a monochromatic path from v to N. We denote by A⁺(u) the set of arcs of D that have u as the initial vertex. We prove that if D is an m-coloured 3-quasitransitive...

Monochromatic paths and monochromatic sets of arcs in bipartite tournaments

Hortensia Galeana-Sánchez, R. Rojas-Monroy, B. Zavala (2009)

Discussiones Mathematicae Graph Theory

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours and all of them are used. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices there is no monochromatic path between them and for every vertex v in V(D)∖N there is a monochromatic path from v to some vertex in N. We denote by A⁺(u) the set of arcs of D that have u as the initial...

Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs

Hortensia Galeana-Sánchez, R. Rojas-Monroy, B. Zavala (2010)

Discussiones Mathematicae Graph Theory

Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively. We call the digraph D an m-coloured digraph if each arc of D is coloured by an element of {1,2,...,m} where m ≥ 1. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if there is no monochromatic path between two vertices of N and if for every vertex v not in N there is a monochromatic path from v to some vertex...

More on betweenness-uniform graphs

Jana Coroničová Hurajová, Tomáš Madaras (2018)

Czechoslovak Mathematical Journal

We study graphs whose vertices possess the same value of betweenness centrality (which is defined as the sum of relative numbers of shortest paths passing through a given vertex). Extending previously known results of S. Gago, J. Hurajová, T. Madaras (2013), we show that, apart of cycles, such graphs cannot contain 2-valent vertices and, moreover, are 3-connected if their diameter is 2. In addition, we prove that the betweenness uniformity is satisfied in a wide graph family of semi-symmetric graphs,...

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