C.N. Campos

Colouring clique-hypergraphs of circulant graphs

Abstract A clique-colouring of a graph G is a colouring of the vertices of G so that no maximal clique of size at least two is monochromatic. The clique-hypergraph, H ( G ) , of a graph G has V ( G ) as its set of vertices and the maximal cliques of G as its hyperedges. A vertex-colouring of H ( G ) is a clique-colouring of G. Determining the clique-chromatic number, the least number for which a graph G admits a clique-colouring, is known to be NP-hard. We establish that the clique-chromatic number for powers of cycles is equal to two, except for odd cycles of size at least five, that need three colours. For odd-seq circulant graphs, we show that their clique-chromatic number is at most four, and determine the cases when it is equal to two.

The total chromatic number of some bipartite graphs

Abstract The total chromatic number χ T ( G ) is the least number of colours needed to colour the vertices and edges of a graph G such that no incident or adjacent elements (vertices or edges) receive the same colour. This work determines the total chromatic number of grids, particular cases of partial grids, near-ladders, and k-dimensional cubes.

Colouring Clique-Hypergraphs of Circulant Graphs

A clique-colouring of a graph G is a colouring of the vertices of G so that no maximal clique of size at least two is monochromatic. The clique-hypergraph,

2K2-Partition Problem☆

{\mathcal{H}(G)}

The 1,2-Conjecture for powers of cycles

, of a graph G has V(G) as its set of vertices and the maximal cliques of G as its hyperedges. A vertex-colouring of

Fulkerson’s Conjecture and Loupekine snarks ☆

{\mathcal{H}(G)}