Plane Graphs with Maximum Degree inline image Are Entirely inline image-Colorable

Authors


  • Contract grant sponsor: Natural Science Foundation of Zhejiang Province, China; Contract grant number: Y6090699 (to Y. W.); Contract grant sponsor: Natural Science Foundation of China; Contract grant number: 10971198 (to Y. W.); Contract grant sponsor: Zhejiang Innovation Project; Contract grant number: T200905 (to Y. W.); Contract grant sponsor: Natural Science Foundation of China; Contract grant number: 11171288 (to Z. M.).

Abstract

Let inline image be a plane graph with the sets of vertices, edges, and faces V, E, and F, respectively. If one can color all elements in inline image using k colors so that any two adjacent or incident elements receive distinct colors, then G is said to be entirely k-colorable. Kronk and Mitchem [Discrete Math 5 (1973) 253-260] conjectured that every plane graph with maximum degree Δ is entirely inline image-colorable. This conjecture has now been settled in Wang and Zhu (J Combin Theory Ser B 101 (2011) 490–501), where the authors asked: is every simple plane graph inline image entirely inline image-colorable? In this article, we prove that every simple plane graph with inline image is entirely inline image-colorable, and conjecture that every simple plane graph, except the tetrahedron, is entirely inline image-colorable.

Ancillary