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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.

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