Discussiones Mathematicae Graph Theory 25(1-2) (2005) 211-215
doi: 10.7151/dmgt.1273

[BIBTex] [PDF] [PS]


Izak Broere and Bonita S. Wilson

Rand Afrikaans University
Johannesburg, Republic of South Africa
e-mail: ib@rau.ac.za

Jozef Bucko

Department of Applied Mathematics
Faculty of Economics, Technical University
B. Nemcovej, 040 01 Košice, Slovak Republic
e-mail: bucko@tuke.sk


Chartrand and Kronk in 1969 showed that there are planar graphs whose vertices cannot be partitioned into two parts inducing acyclic subgraphs. In this note we show that the same is true even in the case when one of the partition classes is required to be triangle-free only.

Keywords: planar graph, hereditary property of graphs, forest and triangle-free graph.

2000 Mathematics Subject Classification: 05C15.


[1] K. Appel and W. Haken, Every planar graph is four colourable, Illinois J. Math. 21 (1977) 429-567.
[2] 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.
[3] M. Borowiecki, I. Broere and P. Mihók, Minimal reducible bounds for planar graphs, Discrete Math. 212 (2000) 19-27, doi: 10.1016/S0012-365X(99)00205-8.
[4] G. Chartrand and H. H. Kronk, The point arboricity of planar graphs, J. London Math. Soc. 44 (1969) 612-616, doi: 10.1112/jlms/s1-44.1.612.
[5] T. Kaiser and R. Skrekovski, Planar graph colorings without short monochromatic cycles, J. Graph Theory 46 (2004) 25-38, doi: 10.1002/jgt.10167.
[6] K. Kuratowski, Sur le problème des courbes gauches en topologie, Fund. Math. 15 (1930) 271-283.
[7] P. Mihók, Minimal reducible bound for outerplanar and planar graphs, Discrete Math. 150 (1996) 431-435, doi: 10.1016/0012-365X(95)00211-E.
[8] C. Thomassen, Decomposing a planar graph into degenerate graphs, J. Combin. Theory (B) 65 (1995) 305-314, doi: 10.1006/jctb.1995.1057.

Received 5 December 2003
Revised 3 November 2004