Abstract
While solving a question on the list coloring of planar graphs, Dvořák and Postle introduced the new notion of DP-coloring (they called it correspondence coloring). A DP-coloring of a graph G reduces the problem of finding a coloring of G from a given list L to the problem of finding a “large” independent set in the auxiliary graph H(G,L) with vertex set {(v, c): v ∈ V (G) and c ∈ L(v)}. It is similar to the old reduction by Plesnevič and Vizing of the k-coloring problem to the problem of finding an independent set of size |V(G)| in the Cartesian product G□K k, but DP-coloring seems more promising and useful than the Plesnevič–Vizing reduction. Some properties of the DP-chromatic number χ DP (G) resemble the properties of the list chromatic number χ l (G) but some differ quite a lot. It is always the case that χ DP (G) ≥ χ l (G). The goal of this note is to introduce DP-colorings for multigraphs and to prove for them an analog of the result of Borodin and Erdős–Rubin–Taylor characterizing the multigraphs that do not admit DP-colorings from some DP-degree-lists. This characterization yields an analog of Gallai’s Theorem on the minimum number of edges in n-vertex graphs critical with respect to DP-coloring.
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The first author was supported by the Illinois Distinguished Fellowship. The second author was supported by the Russian Foundation for Basic Research (Grants 15–01–05867 and 16–01–00499) and the NSF (Grants DMS–1266016 and DMS–1600592).
Urbana–Champaign; Novosibirsk; Barnaul. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 58, No. 1, pp. 36–47, January–February, 2017; DOI: 10.17377/smzh.2017.58.104.
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Bernshteyn, A.Y., Kostochka, A.V. & Pron, S.P. On DP-coloring of graphs and multigraphs. Sib Math J 58, 28–36 (2017). https://doi.org/10.1134/S0037446617010049
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DOI: https://doi.org/10.1134/S0037446617010049