# Generalized total colorings of graphs

• Volume: 31, Issue: 2, page 209-222
• ISSN: 2083-5892

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## Abstract

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An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be additive hereditary properties of graphs. A (P,Q)-total coloring of a simple graph G is a coloring of the vertices V(G) and edges E(G) of G such that for each color i the vertices colored by i induce a subgraph of property P, the edges colored by i induce a subgraph of property Q and incident vertices and edges obtain different colors. In this paper we present some general basic results on (P,Q)-total colorings. We determine the (P,Q)-total chromatic number of paths and cycles and, for specific properties, of complete graphs. Moreover, we prove a compactness theorem for (P,Q)-total colorings.

## How to cite

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Mieczysław Borowiecki, et al. "Generalized total colorings of graphs." Discussiones Mathematicae Graph Theory 31.2 (2011): 209-222. <http://eudml.org/doc/270805>.

@article{MieczysławBorowiecki2011,
abstract = {An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be additive hereditary properties of graphs. A (P,Q)-total coloring of a simple graph G is a coloring of the vertices V(G) and edges E(G) of G such that for each color i the vertices colored by i induce a subgraph of property P, the edges colored by i induce a subgraph of property Q and incident vertices and edges obtain different colors. In this paper we present some general basic results on (P,Q)-total colorings. We determine the (P,Q)-total chromatic number of paths and cycles and, for specific properties, of complete graphs. Moreover, we prove a compactness theorem for (P,Q)-total colorings.},
author = {Mieczysław Borowiecki, Arnfried Kemnitz, Massimiliano Marangio, Peter Mihók},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {hereditary properties; generalized total colorings; paths; cycles; complete graphs},
language = {eng},
number = {2},
pages = {209-222},
title = {Generalized total colorings of graphs},
url = {http://eudml.org/doc/270805},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Mieczysław Borowiecki
AU - Arnfried Kemnitz
AU - Massimiliano Marangio
AU - Peter Mihók
TI - Generalized total colorings of graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 2
SP - 209
EP - 222
AB - An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be additive hereditary properties of graphs. A (P,Q)-total coloring of a simple graph G is a coloring of the vertices V(G) and edges E(G) of G such that for each color i the vertices colored by i induce a subgraph of property P, the edges colored by i induce a subgraph of property Q and incident vertices and edges obtain different colors. In this paper we present some general basic results on (P,Q)-total colorings. We determine the (P,Q)-total chromatic number of paths and cycles and, for specific properties, of complete graphs. Moreover, we prove a compactness theorem for (P,Q)-total colorings.
LA - eng
KW - hereditary properties; generalized total colorings; paths; cycles; complete graphs
UR - http://eudml.org/doc/270805
ER -

## References

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1. [1] M. Borowiecki, I. Broere, M. Frick, P. Mihók and G. Semanišin, A survey of hereditary properties of graphs, Discuss. Math. Graph Theory 17 (1997) 5-50, doi: 10.7151/dmgt.1037. Zbl0902.05026
2. [2] M. Borowiecki and P. Mihók, Hereditary properties of graphs, in: V.R. Kulli (ed.): Advances in Graph Theory, (Vishwa International Publication, Gulbarga, 1991) pp. 42-69.
3. [3] I. Broere, S. Dorfling and E. Jonck, Generalized chromatic numbers and additive hereditary properties of graphs, Discuss. Math. Graph Theory 22 (2002) 259-270, doi: 10.7151/dmgt.1174. Zbl1030.05038
4. [4] R.L. Brooks, On coloring the nodes of a network, Math. Proc. Cambridge Phil. Soc. 37 (1941) 194-197, doi: 10.1017/S030500410002168X. Zbl0027.26403
5. [5] S.A. Burr, An inequality involving the vertex arboricity and edge arboricity of a graph, J. Graph Theory 10 (1986) 403-404, doi: 10.1002/jgt.3190100315. Zbl0651.05030
6. [6] R. Cowen, S.H. Hechler and P. Mihók, Graph coloring compactness theorems equivalent to BPI, Scientia Math. Japonicae 56 (2002) 171-180. Zbl1023.05048
7. [7] G. Chartrand and H.V. Kronk, The point arboricity of planar graphs, J. London Math. Soc. 44 (1969) 612-616, doi: 10.1112/jlms/s1-44.1.612. Zbl0175.50505
8. [8] N.G. de Bruijn and P. Erdös, A colour problem for infinite graphs and a problem in the theory of relations, Indag. Math. 13 (1951) 371-373. Zbl0044.38203
9. [9] M.J. Dorfling and S. Dorfling, Generalized edge-chromatic numbers and additive hereditary properties of graphs, Discuss. Math. Graph Theory 22 (2002) 349-359, doi: 10.7151/dmgt.1180. Zbl1030.05039
10. [10] A. Kemnitz and M. Marangio, [r,s,t] -colorings of graphs, Discrete Math. 307 (2007) 199-207, doi: 10.1016/j.disc.2006.06.030.
11. [11] A. Kemnitz, M. Marangio and P. Mihók, [r,s,t] -chromatic numbers and hereditary properties of graphs, Discrete Math. 307 (2007) 916-922, doi: 10.1016/j.disc.2005.11.055. Zbl1115.05034
12. [12] P. Mihók and G. Semanišin, Unique factorization theorem and formal concept analysis, in: S. Ben Yahia et al. (eds.): Concept Lattices and Their Applications. Fourth International Conference, CLA 2006, Tunis, Tunisia, October 30-November 1, 2006. LNAI 4923. (Springer, Berlin, 2008) pp. 231-238. Zbl1133.05306
13. [13] C.St.J.A. Nash-Williams, Decomposition of finite graphs into forests, J. London Math. Soc. 39 (1964) 12, doi: 10.1112/jlms/s1-39.1.12. Zbl0119.38805
14. [14] V.G. Vizing, On an estimate of the chromatic class of a p-graph (in Russian), Metody Diskret. Analiz. 3 (1964) 25-30.

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