## ARITHMETICALLY MAXIMAL INDEPENDENT SETS IN INFINITE GRAPHS

Stanisław Bylka

Institute of Computer Science
Polish Academy of Sciences
21 Ordona street, 01-237 Warsaw, Poland
e-mail: bylka@ipipan.waw.pl

## Abstract

Families of all sets of independent vertices in graphs are investigated. The problem how to characterize those infinite graphs which have arithmetically maximal independent sets is posed. A positive answer is given to the following classes of infinite graphs: bipartite graphs, line graphs and graphs having locally infinite clique-cover of vertices. Some counter examples are presented.

Keywords: infinite graph, independent set, arithmetical maximal set, line graph.

2000 Mathematics Subject Classification: 05C69, 05C65, 05D05.

## References

 [1] R. Aharoni, König's duality theorem for infinite bipartite graphs, J. London Math. Society 29 (1984) 1-12, doi: 10.1112/jlms/s2-29.1.1. [2] J.C. Bermond and J.C. Meyer, Graphe représentatif des aretes d'un multigraphe, J. Math. Pures Appl. 52 (1973) 229-308. [3] Y. Caro and Z. Tuza, Improved lower bounds on k-independence, J. Graph Theory 15 (1991) 99-107, doi: 10.1002/jgt.3190150110. [4] M.J. Jou and G.J. Chang, Algorithmic aspects of counting independent sets, Ars Combinatoria 65 (2002) 265-277. [5] J. Komar and J. Łoś, König's theorem in the infinite case, Proc. of III Symp. on Operat. Res., Mannheim (1978) 153-155. [6] C.St.J.A. Nash-Williams, Infinite graphs - a survey, J. Combin. Theory 3 (1967) 286-301, doi: 10.1016/S0021-9800(67)80077-2. [7] K.P. Podewski and K. Steffens, Injective choice functions for countable families, J. Combin. Theory (B) 21 (1976) 40-46, doi: 10.1016/0095-8956(76)90025-3. [8] K. Steffens, Matching in countable graphs, Can. J. Math. 29 (1976) 165-168, doi: 10.4153/CJM-1977-016-8. [9] J. Zito, The structure and maximum number of maximum independent sets in trees, J. Graph Theory 15 (1991) 207-221, doi: 10.1002/jgt.3190150208.

Received 28 November 2003
Revised 8 March 2005