The colour theorems of Brooks and Gallai extended

“…In 1994 at a graph theory conference held at Oberwolfach Thomassen pointed out that one can also prove a choosability version of Gallai's result [8], which characterized the subgraphs of k-colour-critical graphs induced by the set of all vertices of degree k − 1. Surprisingly, as Kostochka in [12] has mentioned, this theorem can be easily derived from the result of Borodin [2,3] and Erdős et al [7]. Specifically, Erdős has characterised all d-choosable graphs as follows.…”

Section: Introductionmentioning

confidence: 99%

Time Optimal d-List Colouring of a Graph

Gravin

2010

Computer Science – Theory and Applications

View full text Add to dashboard Cite

Abstract. We present the first linear time algorithm for d-list colouring of a graph-i.e. a proper colouring of each vertex v by colours coming from lists L(v) of sizes at least deg(v). Previously, procedures with such complexity were only known for ∆-list colouring, where for each vertex v one has |L(v)| ≥ ∆, the maximum of the vertex degrees. An implementation of the procedure is available.

show abstract

“…The lower bound in (2) was proved by Gallai [6] for the usual chromatic number, but the proof works for list colouring as well (as was observed, for example, in [10]). Thus the proof of our Theorem 1 also works for list-k-splitting-critical t-uniform hypergraphs.…”

Section: Some Notationmentioning

confidence: 91%

On the Number of Edges in Hypergraphs Critical with Respect to Strong Colourings

Kostochka

Woodall

2000

European Journal of Combinatorics

View full text Add to dashboard Cite

A colouring of the vertices of a hypergraph G is called strong if, for every edge A, the colours of all vertices in A are distinct. It corresponds to a colouring of the generated graph (G) obtained from G by replacing every edge by a clique. We estimate the minimum number of edges possible in a k-critical t-uniform hypergraph with a given number of vertices. In particular we show that, for k ≥ t + 2, the problem reduces in a way to the corresponding problem for graphs. In the case when the generated graph of the hypergraph has bounded clique number, we give a lower bound that is valid for sufficiently large k and is asymptotically tight in k; this bound also holds for list strong colourings.

show abstract

The colour theorems of Brooks and Gallai extended

Cited by 60 publications

References 4 publications

Dirac's map-color theorem for choosability

Dirac's map-color theorem for choosability

Time Optimal d-List Colouring of a Graph

On the Number of Edges in Hypergraphs Critical with Respect to Strong Colourings

Contact Info

Product

Resources

About