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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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##### References:
 [1] Birkhoff, G.D; Lewis, D.C, Chromatic polynomials, Trans. amer. math. soc., 60, 355-451, (1946) · Zbl 0060.41601 [2] Chvatal, V, A note on coefficients of chromatic polynomials, J. combinatorial theory, 9, 95-96, (1970) · Zbl 0203.56603 [3] Hoggar, S.G, Chromatic polynomials and logarithmic concavity, J. combinatorial theory ser. B, 16, 248-254, (1974) · Zbl 0268.05104 [4] Meredith, G.H.J, Coefficients of chromatic polynomials, J. combinatorial theory ser. B, 13, 14-17, (1972) · Zbl 0218.05056 [5] Read, R.C, An introduction to chromatic polynomials, J. combinatorial theory, 4, 52-71, (1968) · Zbl 0173.26203
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