Discussiones Mathematicae Graph Theory 32(3) (2012) 583-602
doi: 10.7151/dmgt.1627

[BIBTex] [PDF] [PS]

Generalized Matrix Graphs and Completely Independent Critical Cliques in any Dimension

John J. Lattanzio and Quan Zheng

Department of Mathematics
Indiana University of Pennsylvania
Indiana, PA 15705, USA

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.

Keywords: matrix graph, chromatic number, critical clique, completely independent critical cliques, double-critical conjecture

2010 Mathematics Subject Classification: 05C15.

References

[1]J. Balogh, A.V. Kostochka, N. Prince, and M. Stiebitz, The Erdös-Lovász Tihany conjecture for quasi-line graphs, Discrete Math., 309 (2009) 3985--3991, doi: 10.1016/j.disc.2008.11.016.
[2]R.A. Brualdi, Introductory Combinatorics, 5th ed, Pearson, (Upper Saddle River, 2010).
[3]G. Chartrand, L. Lesniak, and P. Zhang, Graphs and Digraphs, 5th ed, CRC Press, (Boca Raton, 2010).
[4]G.A. Dirac, A theorem of R.L. Brooks and a conjecture of H. Hadwiger, Proc. Lond. Math. Soc. (3), 7 (1957) 161--195, doi: 10.1112/plms/s3-7.1.161.
[5]G.A. Dirac, The number of edges in critical graphs, J. Reine Angew. Math. 268/269 (1974) 150--164.
[6]P. Erdös, Problems, in: Theory of Graphs, Proc. Colloq., Tihany, (Academic Press, New York, 1968) 361--362.
[7]T.R. Jensen, Dense critical and vertex-critical graphs, Discrete Math. 258 (2002) 63--84, doi: 10.1016/S0012-365X(02)00262-5.
[8]T.R. Jensen and B. Toft, Graph Coloring Problems (Wiley-Interscience, New York, 1995).
[9]K.-i. Kawarabayashi, A.S. Pedersen and B. Toft, Double-critical graphs and complete minors,
Retrieved from http://adsabs.harvard.edu/abs/2008arXiv0810.3133K.
[10]A.V. Kostochka and M. Stiebitz, Colour-critical graphs with few edges, Discrete Math. 191 (1998) 125--137, doi: 10.1016/S0012-365X(98)00100-9.
[11]A.V. Kostochka and M. Stiebitz, On the number of edges in colour-critical graphs and hypergraphs, Combinatorica 20 (2000) 521--530, doi: 10.1007/s004930070005.
[12]J.J. Lattanzio, Completely independent critical cliques, J. Combin. Math. Combin. Comput. 62 (2007) 165--170.
[13]J.J. Lattanzio, Edge double-critical graphs, Journal of Mathematics and Statistics 6 (3) (2010) 357--358, doi: 10.3844/jmssp.2010.357.358.
[14]M. Stiebitz, K5 is the only double-critical 5 -chromatic graph, Discrete Math. 64 (1987) 91--93, doi: 10.1016/0012-365X(87)90242-1.
[15]B. Toft, On the maximal number of edges of critical k -chromatic graphs, Studia Sci. Math. Hungar. 5 (1970) 461--470.

Received 14 February 2011
Revised 9 October 2011
Accepted 15 October 2011