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Uniquely partitionable graphs

Jozef Bucko, Marietjie Frick, Peter Mihók, Roman Vasky (1997)

Discussiones Mathematicae Graph Theory

Let ₁,...,ₙ be properties of graphs. A (₁,...,ₙ)-partition of a graph G is a partition of the vertex set V(G) into subsets V₁, ...,Vₙ such that the subgraph G [ V i ] induced by V i has property i ; i = 1,...,n. A graph G is said to be uniquely (₁, ...,ₙ)-partitionable if G has exactly one (₁,...,ₙ)-partition. A property is called hereditary if every subgraph of every graph with property also has property . If every graph that is a disjoint union of two graphs that have property also has property , then we...

Uniquely partitionable planar graphs with respect to properties having a forbidden tree

Jozef Bucko, Jaroslav Ivančo (1999)

Discussiones Mathematicae Graph Theory

Let ₁, ₂ be graph properties. A vertex (₁,₂)-partition of a graph G is a partition V₁,V₂ of V(G) such that for i = 1,2 the induced subgraph G [ V i ] has the property i . A property ℜ = ₁∘₂ is defined to be the set of all graphs having a vertex (₁,₂)-partition. A graph G ∈ ₁∘₂ is said to be uniquely (₁,₂)-partitionable if G has exactly one vertex (₁,₂)-partition. In this note, we show the existence of uniquely partitionable planar graphs with respect to hereditary additive properties having a forbidden tree....

Unique-Maximum Coloring Of Plane Graphs

Igor Fabrici, Frank Göring (2016)

Discussiones Mathematicae Graph Theory

A unique-maximum k-coloring with respect to faces of a plane graph G is a coloring with colors 1, . . . , k so that, for each face of G, the maximum color occurs exactly once on the vertices of α. We prove that any plane graph is unique-maximum 3-colorable and has a proper unique-maximum coloring with 6 colors.

Unit rectangle visibility graphs.

Dean, Alice M., Ellis-Monaghan, Joanna A., Hamilton, Sarah, Pangborn, Greta (2008)

The Electronic Journal of Combinatorics [electronic only]

Universal tessellations.

David Singerman (1988)

Revista Matemática de la Universidad Complutense de Madrid

All maps of type (m,n) are covered by a universal map M(m,n) which lies on one of the three simply connected Riemann surfaces; in fact M(m,n) covers all maps of type (r,s) where r|m and s|n. In this paper we construct a tessellation M which is universal for all maps on all surfaces. We also consider the tessellation M(8,3) which covers all triangular maps. This coincides with the well-known Farey tessellation and we find many connections between M(8,3) and M.

Universality for and in Induced-Hereditary Graph Properties

Izak Broere, Johannes Heidema (2013)

Discussiones Mathematicae Graph Theory

The well-known Rado graph R is universal in the set of all countable graphs I, since every countable graph is an induced subgraph of R. We study universality in I and, using R, show the existence of 2 א0 pairwise non-isomorphic graphs which are universal in I and denumerably many other universal graphs in I with prescribed attributes. Then we contrast universality for and universality in induced-hereditary properties of graphs and show that the overwhelming majority of induced-hereditary properties...

Universality in Graph Properties with Degree Restrictions

Izak Broere, Johannes Heidema, Peter Mihók (2013)

Discussiones Mathematicae Graph Theory

Rado constructed a (simple) denumerable graph R with the positive integers as vertex set with the following edges: For given m and n with m < n, m is adjacent to n if n has a 1 in the m’th position of its binary expansion. It is well known that R is a universal graph in the set [...] of all countable graphs (since every graph in [...] is isomorphic to an induced subgraph of R). A brief overview of known universality results for some induced-hereditary subsets of [...] is provided. We then construct...

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