Mixing colourings in 2K2-free graphs

Mixing 3-Colourings in Bipartite Graphs

For a 3-colourable graph G, the 3-colour graph of G, denoted C3(G), is the graph with node set the proper vertex 3-colourings of G, and two nodes adjacent whenever the corresponding colourings differ on precisely one vertex of G. We consider the following question : given G, how easily can we decide whether or not C3(G) is connected? We show that the 3-colour graph of a 3-chromatic graph is nev...

Closure of K1 + 2K2-free graphs

Conflict-Free Colourings of Graphs and Hypergraphs

It is shown that, for any > 0, every graph of maximum degree ∆ permits a conflict-free coloring with at most log ∆ colors, that is, a coloring with the property that the neighborhood N(v) of any vertex v, contains an element whose color differs from the color of any other element of N(v). We give an efficient deterministic algorithm to find such a coloring, based on an algorithmic version of th...

On defective colourings of triangle-free graphs

A graph G is (m, k)-colourable if its vertices can be coloured with m colours such that the maximum degree of the subgraph induced on vertices receiving the same colour is at most k. The k-defective chromatic number χk(G) is the least positive integer m for which G is (m, k)-colourable. In 1988 Maddox proved that if either G or Ḡ is triangle-free graph of order p then χk(G)+χk(Ḡ) ≤ 5d p 3k+4e f...

Chromatic bounds for some classes of 2K2-free graphs

A hereditary class G of graphs is χ-bounded if there is a χ-binding function, say f such that χ(G) ≤ f(ω(G)), for every G ∈ G, where χ(G) (ω(G)) denote the chromatic (clique) number of G. It is known that for every 2K2-free graph G, χ(G) ≤ ( ω(G)+1 2 ) , and the class of (2K2, 3K1)-free graphs does not admit a linear χ-binding function. In this paper, we are interested in classes of 2K2-free gr...