On the Absolute Sum of Chromatic Polynomial Coefficient of Graphs

For a century ago, one of the most famous problems in mathematics was to prove the FourColor Problem. During the period that the Four-Color Problem was unsolved, which spanned more than a century, many approaches were introduced with the hopes that they would lead to a solution of this famous problem. In 1913, Birkhoff 1 defined a function P M,λ that gives the number of proper λ-colorings of a map M for a positive integer λ. As we will see, P M,λ is a polynomial in λ for every map M and is called the chromatic polynomial of M. Consequently, if it could be verified that P M, 4 > 0 for every mapM, then this would have established the truth of the Four-Color Conjecture. In 1932, Whitney 2 expanded the study of chromatic polynomials from maps to graphs. While Whitney obtained a number of results on chromatic polynomials of graphs and others obtained results on the roots of chromatic polynomials of planar graphs, this did not contribute to a proof of the Four-Color Conjecture. Renewed interest in chromatic polynomials of graphs occurred in 1968 when Read 3 wrote a survey paper on chromatic polynomials. Let G be a graph and λ ∈ . A mapping f : V G → {1, 2, . . . , λ} is called a λ-coloring of G if f u / f v whenever the vertices u and v are adjacent in G. For a positive integer λ, the number of different proper λ-colorings of G is denoted by P G, λ and is called the chromatic polynomial of G. By convention, P G, 0 0, and P G, λ ≥ 1 if and only if G is λ-colorable. More precisely, we have