Abstract
As starting point, we formulate a corollary to the Quantitative Combinatorial Nullstellensatz. This corollary does not require the consideration of any coefficients of polynomials, only evaluations of polynomial functions. In certain situations, our corollary is more directly applicable and more ready-to-go than the Combinatorial Nullstellensatz itself. It is also of interest from a numerical point of view. We use it to explain a well-known connection between the sign of 1-factorizations (edge colorings) and the List Edge Coloring Conjecture. For efficient calculations and a better understanding of the sign, we then introduce and characterize the sign of single 1-factors. We show that the product over all signs of all the 1-factors in a 1-factorization is the sign of that 1-factorization. Using this result in an algorithm, we attempt to prove the List Edge Coloring Conjecture for all graphs with up to 10 vertices. This leaves us with some exceptional cases that need to be attacked with other methods.
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Alon, N.: Restricted colorings of graphs. In: Surveys in Combinatorics. London Mathematical Society Lecture Notes Series, vol. 187, pp. 1–33. Cambridge University Press, Cambridge (1993)
Alon, N.: Combinatorial nullstellensatz. Comb. Probab. Comput. 8(1–2), 7–29 (1999)
Ellingham, M.N., Goddyn, L.: List edge colourings of some 1-factorable multigraphs. Combinatorica 16, 343–352 (1996)
Galvin, F.: The list chromatic index of a bipartite multigraph. J. Comb. Theory Ser. B 63, 153–158 (1995)
Häggkvist, R., Janssen, J.: New bounds on the list-chromatic index of the complete graph and other simple graphs. Comb. Probab. Comput. 6, 295–313 (1997)
Jensen, T.R., Toft, B.: Graph Coloring Problems. Wiley, New York (1995)
Kahn, J.: Asymptotically good list-colorings. J. Comb. Theory Ser. A 73(1), 1–59 (1996)
Meringer, M.: Connected regular graphs. http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html
Meringer, M.: Fast generation of regular graphs and construction of cages. J. Gr. Theory 30, 137–146 (1999)
Petersen, J.: Die theorie der regularen graphs. Acta Math. 15, 193–220 (1891)
SageMath: The sage mathematics software system (version 7.4.1). The Sage Developers (2017). http://www.sagemath.org
Schauz, U.: Algebraically solvable problems: describing polynomials as equivalent to explicit solutions. Electron. J. Comb. 15, R10 (2008)
Schauz, U.: Mr. Paint and Mrs. Correct. Electron. J. Comb. 15, R145 (2008)
Schauz, U.: A paintability version of the combinatorial Nullstellensatz, and list colorings of \(k\)-partite \(k\)-uniform hypergraphs. Electron. J. Comb. 17(1), R176 (2010)
Schauz, U.: Proof of the list edge coloring conjecture for complete graphs of prime degree. Electron. J. Comb. 21(3), 3–43 (2014)
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Schauz, U. (2018). Orientations of 1-Factors and the List Edge Coloring Conjecture. In: Brankovic, L., Ryan, J., Smyth, W. (eds) Combinatorial Algorithms. IWOCA 2017. Lecture Notes in Computer Science(), vol 10765. Springer, Cham. https://doi.org/10.1007/978-3-319-78825-8_19
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DOI: https://doi.org/10.1007/978-3-319-78825-8_19
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