Maximal graphs with respect to hereditary properties

• Autorzy: Izak Broere , Marietjie Frick , Gabriel Semanišin
• Języki publikacji: EN

Abstrakty

EN

A property of graphs is a non-empty set of graphs. A property P is called hereditary if every subgraph of any graph with property P also has property P. Let P₁, ...,Pₙ be properties of graphs. We say that a graph G has property P₁∘...∘Pₙ if the vertex set of G can be partitioned into n sets V₁, ...,Vₙ such that the subgraph of G induced by V_i has property $P_i$; i = 1,..., n. A hereditary property R is said to be reducible if there exist two hereditary properties P₁ and P₂ such that R = P₁∘P₂. If P is a hereditary property, then a graph G is called P- maximal if G has property P but G+e does not have property P for every e ∈ E([G̅]). We present some general results on maximal graphs and also investigate P-maximal graphs for various specific choices of P, including reducible hereditary properties.

Słowa kluczowe

EN

hereditary property of graphs; maximal graphs; vertex partition

Kategorie tematyczne

• 05C15: Coloring of graphs and hypergraphs

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 1997
• Tom: 17
• Numer: 1
• Strony: 51-66

Daty

wydano

1997

otrzymano

1997-03-13

Twórcy

autor

Izak Broere

• Department of Mathematics, Rand Afrikaans University, P.O. Box 524, Auckland Park, 2006 South Africa

autor

Marietjie Frick

• Department of Mathematics, Applied Mathematics and Astronomy, University of South Africa, P.O. Box 392, Pretoria, 0001 South Africa

autor

Gabriel Semanišin

• Department of Geometry and Algebra, P.J.Šafárik University, 041 54 Košice, Slovak Republic

Bibliografia

• [1] G. Benadé, I. Broere, B. Jonck and M. Frick, Uniquely $(m,k)^τ$-colourable graphs and k-τ-saturated graphs, Discrete Math. 162 (1996) 13-22, doi: 10.1016/0012-365X(95)00301-C.
• [2] M. Borowiecki, I. Broere and P. Mihók, Minimal reducible bounds for planar graphs, submitted.
• [3] M. Borowiecki, J. Ivančo, P. Mihók and G. Semanišin, Sequences realizable by maximal k-degenerate graphs, J. Graph Theory 19 (1995) 117-124; MR96e:05078.
• [4] M. Borowiecki and P. Mihók, Hereditary properties of graphs, in: V.R. Kulli, ed., Advances in Graph Theory (Vishwa International Publication, Gulbarga, 1991) 41-68.
• [5] I. Broere, M. Frick and P. Mihók, On the order of uniquely partitionable graphs, submitted.
• [6] G. Chartrand and L. Lesniak, Graphs and Digraphs, (Wadsworth & Brooks/Cole, Monterey California, 1986).
• [7] P. Erdős and T. Gallai, On the minimal number of vertices representing the edges of a graph, Magyar Tud. Akad. Math. Kutató Int. Közl. 6 (1961) 181-203; MR 26#1878.
• [8] Z. Filáková, P. Mihók and G. Semanišin, On maximal k-degenerate graphs, to appear in Math. Slovaca.
• [9] A. Hajnal, A theorem on k-saturated graphs, Canad. J. Math. 17 (1965) 720-724; MR31#3354.
• [10] L. Kászonyi and Zs. Tuza, Saturated graphs with minimal number of edges, J. Graph Theory 10 (1986) 203-210, doi: 10.1002/jgt.3190100209.
• [11] J. Kratochvíl, P. Mihók and G. Semanišin, Graphs maximal with respect to hom-properties, Discussiones Mathematicae Graph Theory 17 (1997) 77-88, doi: 10.7151/dmgt.1040.
• [12] R. Lick and A.T. White, k-degenerate graphs, Canad. J. Math. 22 (1970) 1082-1096; MR42#1715.
• [13] P. Mihók, Additive hereditary properties and uniquely partitionable graphs, in: M. Borowiecki and Z. Skupień, eds., Graphs, Hypergraphs and Matroids (Zielona Góra, 1985) 49-58.
• [14] P. Mihók and G. Semanišin, Reducible properties of graphs, Discussiones Math. Graph Theory 15 (1995) 11-18; MR96c:05149, doi: 10.7151/dmgt.1002.
• [15] J. Mitchem, An extension of Brooks' theorem to r-degenerate graphs, Discrete Math. 17 (1977) 291-298; MR 55#12561.
• [16] J. Mitchem, Maximal k-degenerate graphs, Utilitas Math. 11 (1977) 101-106.
• [17] G. Semanišin, On some variations of extremal graph problems, Discussiones Mathematicae Graph Theory 17 (1997) 67-76, doi: 10.7151/dmgt.1039.
• [18] J.M.S. Simoes-Pereira, On graphs uniquely partitionable into n-degenerate subgraphs, in: Infinite and finite sets - Colloquia Math. Soc. J. Bólyai 10 (North-Holland Amsterdam, 1975) 1351-1364; MR53#2758.
• [19] J.M.S. Simoes-Pereira, A survey on k-degenerate graphs, Graph Theory Newsletter 5 (6) (75/76) 1-7; MR 55#199.

Identyfikatory

DOI

10.7151/dmgt.1038