Combinatorial Properties of the Family of Maximum Stable Sets of a Graph
Abstract
The stability number alpha(G) of a graph G is the cardinality of a maximum stable set in G, xi(G) denotes the size of core(G), where core(G) is the intersection of all maximum stable sets of G. In this paper we prove that for a graph G without isolated vertices, the following assertions are true: (i) if xi(G)< 2, then G is quasiregularizable; (ii) if G is of order n and alpha(G) > (n+k1)/2, for some k > 0, then xi(G) > k, and xi(G) > k+1, whenever n+k1 is even. The last finding is a strengthening of a result of Hammer, Hansen, and Simeone, which states that alpha(G) > n/2 implies xi(G) > 0. G is a KoenigEgervary graph if n equals the sum of its stability number and the cardinality of a maximum matching. For KoenigEgervary graphs, we prove that alpha(G) > n/2 holds if and only if xi(G) is greater than the size of the neighborhood of core(G). Moreover, for bipartite graphs without isolated vertices, alpha(G) > n/2 is equivalent to xi(G) > 1. We also show that Hall's marriage Theorem is valid for KoenigEgervary graphs, and it is sufficient to check Hall's condition only for one specific stable set, namely, for core(G).
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 1999
 arXiv:
 arXiv:math/9912047
 Bibcode:
 1999math.....12047L
 Keywords:

 Combinatorics;
 05C69 (Primary) 05C70 (Secondary)
 EPrint:
 13 pages