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\(\sigma\)-polynomials and graph coloring. (English) Zbl 0845.05043
Let \(G\) be a graph, and let \(P(G, \lambda)\) be the chromatic polynomial of \(G\). A well-known decomposition of \(G\) leads to the expression of \(P(G, \lambda)\) in terms of chromatic polynomials of complete graphs \(K_i\). The author uses the same decomposition to obtain a new graph polynomial, which he calls \(\sigma (G)\), the \(\sigma\)-polynomial of \(G\). The coefficients of \(\sigma (G)\) are the nonzero coefficients in the complete-graph decomposition of \(P (G, \lambda)\). If \(P_i = P (K_i, \lambda)\), and if \(i_0\) is the smallest integer \(i\) such that \(P (K_i, \lambda)\) has a nonzero coefficient in the complete-graph expansion of \(P (G, \lambda)\), then \(\sigma (G) = (1/P_{i_0}) P (G, \lambda)\). The author gives tight bounds on the coefficients of \(\sigma\)-polynomials and derives these polynomials for several classes of graphs.

MSC:
05C15 Coloring of graphs and hypergraphs
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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