# Hamiltonicity and Generalised Total Colourings of Planar Graphs

Mieczysław Borowiecki; Izak Broere

Discussiones Mathematicae Graph Theory (2016)

- Volume: 36, Issue: 2, page 243-257
- ISSN: 2083-5892

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topMieczysław Borowiecki, and Izak Broere. "Hamiltonicity and Generalised Total Colourings of Planar Graphs." Discussiones Mathematicae Graph Theory 36.2 (2016): 243-257. <http://eudml.org/doc/277131>.

@article{MieczysławBorowiecki2016,

abstract = {The total generalised colourings considered in this paper are colourings of graphs such that the vertices and edges of the graph which receive the same colour induce subgraphs from two prescribed hereditary graph properties while incident elements receive different colours. The associated total chromatic number is the least number of colours with which this is possible. We study such colourings for sets of planar graphs and determine, in particular, upper bounds for these chromatic numbers for proper colourings of the vertices while the monochromatic edge sets are allowed to be forests. We also prove that if an even planar triangulation has a Hamilton cycle H for which there is no cycle among the edges inside H, then such a graph needs at most four colours for a total colouring as described above. The paper is concluded with some conjectures and open problems.},

author = {Mieczysław Borowiecki, Izak Broere},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {even planar triangulation; total colouring; Hamilton cycle; hereditary property},

language = {eng},

number = {2},

pages = {243-257},

title = {Hamiltonicity and Generalised Total Colourings of Planar Graphs},

url = {http://eudml.org/doc/277131},

volume = {36},

year = {2016},

}

TY - JOUR

AU - Mieczysław Borowiecki

AU - Izak Broere

TI - Hamiltonicity and Generalised Total Colourings of Planar Graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2016

VL - 36

IS - 2

SP - 243

EP - 257

AB - The total generalised colourings considered in this paper are colourings of graphs such that the vertices and edges of the graph which receive the same colour induce subgraphs from two prescribed hereditary graph properties while incident elements receive different colours. The associated total chromatic number is the least number of colours with which this is possible. We study such colourings for sets of planar graphs and determine, in particular, upper bounds for these chromatic numbers for proper colourings of the vertices while the monochromatic edge sets are allowed to be forests. We also prove that if an even planar triangulation has a Hamilton cycle H for which there is no cycle among the edges inside H, then such a graph needs at most four colours for a total colouring as described above. The paper is concluded with some conjectures and open problems.

LA - eng

KW - even planar triangulation; total colouring; Hamilton cycle; hereditary property

UR - http://eudml.org/doc/277131

ER -

## References

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