Graph polynomials and group coloring of graphs
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hdl:2117/371592
Tipus de documentArticle
Data publicació2022-05-01
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Abstract
Let A be an Abelian group and let G be a simple graph. We say that G is A-colorable if, for some fixed orientation of G and every edge labeling l:E(G)¿A, there exists a vertex coloring c by the elements of A such that c(y)-c(x)¿l(e), for every edge e=xy (oriented from x to y). Given an arbitrary field F, suppose that each edge e=xy of G is assigned a triple (ae,be,ce)¿F3, with ae,be¿0. We say that G is F-colorable if for every such edge labeling there exists a vertex coloring f by the elements of F such that aef(x)+bef(y)+ce¿0, for every edge e=xy. Clearly, F-colorability of a graph implies its A-colorability, where A is the additive group of the field F. A graph G is said to be F-k-choosable if it is F-colorable from arbitrary lists of elements of F, each of size k, assigned to the vertices. Recently, R. Langhede and C. Thomassen [Discrete Math. 344 (2021), no. 9, Paper No. 112474; MR4268692] proved that every simple planar graph on n vertices is Z5-colorable and moreover it has at least 2n/9 different Z5-colorings. In the paper under review, using a different approach based on graph polynomials, the authors extend this result to K5-minor-free graphs in the more general setting of field coloring. More specifically, they prove that every such graph on n vertices is F-5-choosable, whenever F is an arbitrary field with at least 5 elements. Moreover, the number of colorings (for every list assignment) is at least 5n/4.
CitacióBosek, B. [et al.]. Graph polynomials and group coloring of graphs. "European journal of combinatorics", 1 Maig 2022, vol. 102, núm. Article 103505.
ISSN0195-6698
Versió de l'editorhttps://www.sciencedirect.com/science/article/pii/S0195669821001992
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