In this paper a concept 𝓠-Ramsey Class of graphs is introduced, where 𝓠 is a class of bipartite graphs. It is a generalization of well-known concept of Ramsey Class of graphs. Some 𝓠-Ramsey Classes of graphs are presented (Theorem 1 and 2). We proved that 𝓣₂, the class of all outerplanar graphs, is not 𝓓₁-Ramsey Class (Theorem 3). This results leads us to the concept of acyclic reducible bounds for a hereditary property 𝓟 . For 𝓣₂ we found two bounds (Theorem 4). An improvement, in some sense, of that in Theorem is given.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1330,
author = {Mieczys\l aw Borowiecki and Anna Fiedorowicz},
title = {On partitions of hereditary properties of graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {26},
year = {2006},
pages = {377-387},
zbl = {1139.05018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1330}
}
Mieczysław Borowiecki; Anna Fiedorowicz. On partitions of hereditary properties of graphs. Discussiones Mathematicae Graph Theory, Tome 26 (2006) pp. 377-387. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1330/
[000] [1] P. Boiron, E. Sopena and L. Vignal, Acyclic improper colorings of graphs, J. Graph Theory 32 (1999) 97-107, doi: 10.1002/(SICI)1097-0118(199909)32:1<97::AID-JGT9>3.0.CO;2-O | Zbl 0929.05031
[001] [2] P. Boiron, E. Sopena and L. Vignal, Acyclic improper colourings of graphs with bounded degree, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 49 (1999) 1-9. | Zbl 0930.05042
[002] [3] O.V. Borodin, On acyclic colorings of planar graphs, Discrete Math. 25 (1979) 211-236, doi: 10.1016/0012-365X(79)90077-3. | Zbl 0406.05031
[003] [4] O.V. Borodin, A.V. Kostochka, A. Raspaud and E. Sopena, Acyclic colourings of 1-planar graphs, Discrete Applied Math. 114 (2001) 29-41, doi: 10.1016/S0166-218X(00)00359-0. | Zbl 0996.05053
[004] [5] O.V. Borodin, A.V. Kostochka and D.R. Woodall, Acyclic colorings of planar graphs with large girth, J. London Math. Soc. 60 (1999) 344-352, doi: 10.1112/S0024610799007942. | Zbl 0940.05032
[005] [6] M. Borowiecki, I. Broere, M. Frick, P. Mihók and G. Semanišin, A survey of hereditary properties of graphs, Discussiones Mathematicae Graph Theory 17 (1997) 5-50, doi: 10.7151/dmgt.1037. | Zbl 0902.05026
[006] [7] M.I. Burstein, Every 4-valent graph has an acyclic 5-coloring, Soobsc. Akad. Gruzin. SSR 93 (1979) 21-24 (in Russian). | Zbl 0397.05023
[007] [8] G. Ding, B. Oporowski, D.P. Sanders and D. Vertigan, Partitioning graphs of bounded tree-width, Combinatorica 18 (1998) 1-12, doi: 10.1007/s004930050001. | Zbl 0924.05022
[008] [9] R. Diestel, Graph Theory (Springer, Berlin, 1997).
[009] [10] B. Grunbaum, Acyclic coloring of planar graphs, Israel J. Math. 14 (1973) 390-412, doi: 10.1007/BF02764716. | Zbl 0265.05103
[010] [11] P. Mihók and G. Semanišin, Reducible properties of graphs, Discuss. Math. Graph Theory 15 (1995) 11-18, doi: 10.7151/dmgt.1002. | Zbl 0829.05057