Generalized matrix graphs and completely independent critical cliques in any dimension

John J. Lattanzio; Quan Zheng

Discussiones Mathematicae Graph Theory (2012)

  • Volume: 32, Issue: 3, page 583-602
  • ISSN: 2083-5892

Abstract

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For natural numbers k and n, where 2 ≤ k ≤ n, the vertices of a graph are labeled using the elements of the k-fold Cartesian product Iₙ × Iₙ × ... × Iₙ. Two particular graph constructions will be given and the graphs so constructed are called generalized matrix graphs. Properties of generalized matrix graphs are determined and their application to completely independent critical cliques is investigated. It is shown that there exists a vertex critical graph which admits a family of k completely independent critical cliques for any k, where k ≥ 2. Some attention is given to this application and its relationship with the double-critical conjecture that the only vertex double-critical graph is the complete graph.

How to cite

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John J. Lattanzio, and Quan Zheng. "Generalized matrix graphs and completely independent critical cliques in any dimension." Discussiones Mathematicae Graph Theory 32.3 (2012): 583-602. <http://eudml.org/doc/271081>.

@article{JohnJ2012,
abstract = {For natural numbers k and n, where 2 ≤ k ≤ n, the vertices of a graph are labeled using the elements of the k-fold Cartesian product Iₙ × Iₙ × ... × Iₙ. Two particular graph constructions will be given and the graphs so constructed are called generalized matrix graphs. Properties of generalized matrix graphs are determined and their application to completely independent critical cliques is investigated. It is shown that there exists a vertex critical graph which admits a family of k completely independent critical cliques for any k, where k ≥ 2. Some attention is given to this application and its relationship with the double-critical conjecture that the only vertex double-critical graph is the complete graph.},
author = {John J. Lattanzio, Quan Zheng},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {matrix graph; chromatic number; critical clique; completely independent critical cliques; double-critical conjecture; generalized matrix graph},
language = {eng},
number = {3},
pages = {583-602},
title = {Generalized matrix graphs and completely independent critical cliques in any dimension},
url = {http://eudml.org/doc/271081},
volume = {32},
year = {2012},
}

TY - JOUR
AU - John J. Lattanzio
AU - Quan Zheng
TI - Generalized matrix graphs and completely independent critical cliques in any dimension
JO - Discussiones Mathematicae Graph Theory
PY - 2012
VL - 32
IS - 3
SP - 583
EP - 602
AB - For natural numbers k and n, where 2 ≤ k ≤ n, the vertices of a graph are labeled using the elements of the k-fold Cartesian product Iₙ × Iₙ × ... × Iₙ. Two particular graph constructions will be given and the graphs so constructed are called generalized matrix graphs. Properties of generalized matrix graphs are determined and their application to completely independent critical cliques is investigated. It is shown that there exists a vertex critical graph which admits a family of k completely independent critical cliques for any k, where k ≥ 2. Some attention is given to this application and its relationship with the double-critical conjecture that the only vertex double-critical graph is the complete graph.
LA - eng
KW - matrix graph; chromatic number; critical clique; completely independent critical cliques; double-critical conjecture; generalized matrix graph
UR - http://eudml.org/doc/271081
ER -

References

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  15. [15] B. Toft, On the maximal number of edges of critical k -chromatic graphs, Studia Sci. Math. Hungar. 5 (1970) 461-470. Zbl0228.05111

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