Unsolved graph colouring problems

A general upper bound for the cyclic chromatic number of 3‐connected plane graphs

“…Roughly speaking, a consequence of Euler's polyhedral formula enables us to show that any 3-connected plane graph, having maximum face degree * ∈ [9,14], and at least * +6 vertices, contains a ( * , 5)-reducible configuration. Thus, there is no 5-minimal 3-connected plane graph, which establishes the desired upper bound for the cyclic chromatic number of 3-connected plane graphs.…”

“…From Table I we see that it is sufficient to deal with the case of 3-connected plane graphs with maximum face degree in [9,14]. If H is a 3-connected plane graph with * (H ) ∈ [9,14] and To formulate the Redistribution Rules we need some preparatory notions. A 3-vertex is of type…”

“…If y j is a special 4-vertex, then, by (14) and (15), c 5 ( Let x be a 3-vertex not incident with a 3-face. If x is of type (4,7,9), consider a 9-face f incident with x.…”

The unique factorization theorem

“…have been stated in the book [8] of Jensen and Toft "Graph Coloring Problems". Partial results for some subclasses of inducedhereditary properties may be found in [11,12,9,13].…”

Time Optimal d-List Colouring of a Graph

“…Recently this line of research became even more popular than simple colourings. For example it is studied in depth in [9,18,13,11] and many other sources, e.g. as recently as [10].…”