5-list-coloring Planar Graphs with Distant Precolored Vertices

نویسندگان

  • Zdenek Dvorak
  • Bernard Lidický
  • Bojan Mohar
  • Luke Postle
چکیده

We prove the conjecture of Albertson stating that every planar graph can be 5-list-colored, even if it contains precolored vertices, as long as they are sufficiently far apart from each other. In order to prove this claim, we also give bounds on the sizes of graphs critical with respect to 5-list coloring. In particular, if G is a planar graph, H is a connected subgraph of G and L is an assignment of lists of colors to the vertices of G such that |L(v)| ≥ 5 for every v ∈ V (G) \ V (H) and G is not L-colorable, then G contains a subgraph with O(|H|2) vertices that is not L-colorable. 1 List colorings of planar graphs For a graph G, a list assignment is a function L that assigns a set of colors to each vertex of G. For v ∈ V (G), we say that L(v) is the list of v. An ∗Computer Science Institute of Charles University, Prague, Czech Republic. E-mail: [email protected]. Supported by Institute for Theoretical Computer Science (ITI), project 1M0545 of Ministry of Education of Czech Republic, and by project GA201/09/0197 (Graph colorings and flows: structure and applications) of Czech Science Foundation. †Charles University, Prague, Czech Republic and University of Illinois at UrbanaChapmaign, Urbana, USA E-mail: [email protected]. ‡Department of Mathematics, Simon Fraser University, Burnaby, B.C. V5A 1S6. Email: [email protected]. Supported in part by an NSERC Discovery Grant (Canada), by the Canada Research Chair program, and by the Research Grant P1–0297 of ARRS (Slovenia). §On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia. ¶School of Mathematics, Georgia Tech, Atlanta, GA.

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عنوان ژورنال:
  • J. Comb. Theory, Ser. B

دوره 122  شماره 

صفحات  -

تاریخ انتشار 2017