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Lower bounds on the stability number of graphs computed in terms of degrees. (English) Zbl 0755.05055

Summary: V. K. Wei discovered that the stability number, \(\alpha(G)\), of a graph, \(G\), with degree sequence \(d_ 1\), \(d_ 2,\dots,d_ n\) is at least \[ w(G)=\sum^ n_{i=1}{1\over d_ i+1}. \] It is shown that this bound can be replaced by a function \(b(G)\), computable from \(d_ 1\), \(d_ 2,\dots,d_ n\) using only \(O(n)\) additions and comparisons. For all graphs, \(b(G)\geq\lceil w(G)\rceil\), the inequality sometimes holding as strict. In addition, it is shown that Wei’s bound can be increased by \((w(G)-1)/\Delta(\Delta+1)\) when \(G\) is connected, by \(w(G)k/2\Delta(\Delta+1)\) when \(G\) is \(k\)-connected but not complete, and by \((w(G)+m-n)/\Delta(\Delta+1)\) when \(G\) is triangle-free; in each case, \(\Delta,n\), and \(m\) denote the largest degree of a vertex in \(G\), the number of vertices of \(G\), and the number of edges of \(G\), respectively.

MSC:

05C35 Extremal problems in graph theory
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References:

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