Consider the following two ways to colour the vertices of a graph where the requirement that
adjacent vertices get distinct colours is relaxed. A colouring has" defect" $ d $ if each …

This paper exploits the remarkable new method of Galvin (J. Combin. Theory Ser. B63
(1995), 153–158), who proved that the list edge chromatic numberχ′ list (G) of a bipartite …

After a brief historical account, a few simple structural theorems about plane graphs useful
for coloring are stated, and two simple applications of discharging are given. Afterwards, the …

We survey the literature on those variants of the {\em chromatic number\/} problem where not
only a proper coloring has to be found (ie, adjacent vertices must not receive the same color) …

The total chromatic number of any multigraph with maximum degree five is at most seven
Page 1 DISCRETE MATHEMATICS ELSEVIER Discrete Mathematics 162 (1996) 199-214 …

In 1965 Ringel raised a 6 color problem for graphs that can be stated in at least three
different forms. In particular, is it possible to color the vertices and faces of every plane graph …

Given a graph G, a total k‐coloring of G is a simultaneous coloring of the vertices and edges
of G with at most k colors. If Δ (G) is the maximum degree of G, then no graph has a total Δ …

It is well known that every planar graph contains a vertex of degree at most 5. A theorem of
Kotzig states that every 3-connected planar graph contains an edge whose endvertices …

We provide a “how-to” guide to the use and application of the Discharging Method. Our aim
is not to exhaustively survey results proved by this technique, but rather to demystify the …

Total colorings of planar graphs with large maximum degree Page 1 Total Colorings of Planar
Graphs with Large Maximum Degree OV Borodin,1,* AV Kostochka,2,† and DR Woodall3 1 …