The total-chromatic number of some families of snarks

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)The total-chromatic number chi(T) (C) 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. It is known that the problem of determining the total-chromatic number is NP-hard, and it remains NP-hard even for cubic bipartite graphs. Snarks are simple connected bridgeless cubic graphs that are not 3-edge-colourable. In this paper, we show that the total-chromatic number is 4 for three infinite families of snarks, namely, the Flower Snarks. the Goldberg Snarks, and the Twisted Goldberg Snarks. This result reinforces the conjecture that all snarks have total-chromatic number 4. Moreover, we give recursive procedures to construct a total-colouring that uses 4 colours in each case. (C) 2011 Elsevier B.V. All rights reserved.31112984988Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP

Colouring Clique-Hypergraphs of Circulant Graphs

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)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, , 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 is a clique-colouring of G. Determining the clique-chromatic number, the least number of colours for which a graph G admits a clique-colouring, is known to be NP-hard. In this work, we establish that the clique-chromatic number of 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. Similar bounds for the chromatic number of these graphs are also obtained.29617131720Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP