List Colorings of K5-Minor-Free Graphs With Special List Assignments

  • Authors:

  • Daniel W. Cranston;Anja Pruchnewski;Zsolt Tuza;Margit Voigt

  • Affiliations:

  • Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, Virginia;Faculty of Mathematics and Natural Sciences, Ilmenau University of Technology, Ilmenau, Germany;Computer and Automation Institute, Hungarian Academy of Sciences, Budapest, Hungary and Department of Computer Science and Systems Technology, University Of Pannonia, Veszprém, Hungary;Faculty of Information Technology and Mathematics, University of Applied Sciences, Dresden, Germany

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2012

Quantified Score

Hi-index 0.00

Visualization

Abstract

The following question was raised by Bruce Richter. Let G be a planar, 3-connected graph that is not a complete graph. Denoting by d(v) the degree of vertex v, is G L-list colorable for every list assignment L with |L(v)| = min{d(v), 6} for all v∈V(G)? More generally, we ask for which pairs (r, k) the following question has an affirmative answer. Let r and k be the integers and let G be a K5-minor-free r-connected graph that is not a Gallai tree (i.e. at least one block of G is neither a complete graph nor an odd cycle). Is G L-list colorable for every list assignment L with |L(v)| = min{d(v), k} for all v∈V(G)? We investigate this question by considering the components of G[Sk], where Sk: = {v∈V(G)|d(v)8k} is the set of vertices with small degree in G. We are especially interested in the minimum distance d(Sk) in G between the components of G[Sk]. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:18–30, 2012 © 2012 Wiley Periodicals, Inc.