C.P. de Mello

Edge-colouring of join graphs

A join graph is the complete union of two arbitrary graphs. We give sufficient conditions for a join graph to be 1-factorizable. As a consequence of our results, the Hiltons Overfull Subgraph Conjecture holds true for several subclasses of join graphs.

The clique operator on graphs with few P 4 's

The clique graph of a graph G is the intersection graph K (G) of the (maximal) cliques of G. The iterated clique graphs Kn (G) are defined by KO(G) = G and Ki(G) = K(Ki-1(G)), i > 0 and K is the clique operator. In this article we use the modular decomposition technique to characterize the K-behaviour of some classes of graphs with few P4s.These characterizations lead to polynomial time algorithms for deciding the K-convergence or K-divergence of any graph in the class.

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.

The total chromatic number of split-indifference graphs ☆

Abstract The total chromatic number of a graph G , χ T ( G ) , is the least number of colours sufficient to colour the vertices and edges of a graph such that no incident or adjacent elements (vertices or edges) receive the same colour. The Total Colouring Conjecture (TCC) states that every simple graph G has χ T ( G ) ≤ Δ ( G ) + 2 , and it is a challenging open problem in Graph Theory. For both split graphs and indifference graphs, the TCC holds, and χ T ( G ) = Δ ( G ) + 1 when Δ ( G ) is even. For a split-indifference graph G with odd Δ ( G ) , we give conditions for its total chromatic number to be Δ ( G ) + 2 , and we build a ( Δ ( G ) + 1 ) -total colouring otherwise. Also, we pose a conjecture for a class of graphs that generalizes split-indifference graphs.

The P4-sparse Graph Sandwich Problem☆

Abstract The P 4 -sparse Graph Sandwich Problem asks, given two graphs G 1 = ( V , E 1 ) and G 2 = ( V , E 2 ) , whether there exists a graph G = ( V , E ) such that E 1 ⊆ E ⊆ E 2 and G is P 4 -sparse. In this paper we present a polynomial-time algorithm for solving the Graph Sandwich Problem for P 4 -sparse graphs.

AVD-total-chromatic number of some families of graphs with Δ ( G ) = 3

An AVD-total-colouring of a simple graph G is a mapping π : V ( G ) ź E ( G ) ź { 1 , ź , k } , k ź 1 , such that: (i) for each pair of adjacent or incident elements x , y ź V ( G ) ź E ( G ) , π ( x ) ź π ( y ) ; and (ii) for each pair of adjacent vertices x , y ź V ( G ) , sets { π ( x ) } ź { π ( x v ) : x v ź E ( G ) , v ź V ( G ) } and { π ( y ) } ź { π ( y v ) : y v ź E ( G ) , v ź V ( G ) } are distinct. The AVD-total-chromatic number, ź a ź ( G ) , is the smallest number of colours for which G admits an AVD-total-colouring. In 2010, J.źHulgan conjectured that any simple graph G with maximum degree three has ź a ź ( G ) ź 5 . In this article, we verify Hulgans Conjecture for simple graphs G with Δ ( G ) = 3 and without adjacent vertices of maximum degree, and also for the following families of snarks: the flower snarks, generalized Blanusa snarks, and L P 1 -snarks. In fact, we determine the exact value of ź a ź ( G ) for all families considered in this work.

AVD-total-colouring of complete equipartite graphs

An AVD-total-colouring of a simple graph G is a mapping π : V ( G ) ? E ( G ) ? C , C a set of colours, such that: (i) for each pair of adjacent or incident elements x , y ? V ( G ) ? E ( G ) , π ( x ) ? π ( y ) ; (ii) for each pair of adjacent vertices x , y ? V ( G ) , sets { π ( x ) } ? { π ( x v ) : x v ? E ( G ) , v ? V ( G ) } and { π ( y ) } ? { π ( y v ) : y v ? E ( G ) , v ? V ( G ) } are distinct. The AVD-total-chromatic number, ? a ? ( G ) , is the smallest number of colours for which G admits an AVD-total-colouring. In 2005, Zhang et?al. conjectured that ? a ? ( G ) ? Δ ( G ) + 3 for any simple graph G . In this article this conjecture is verified for any complete equipartite graph. Moreover, if G is a complete equipartite graph of even order, then it is shown that ? a ? ( G ) = Δ ( G ) + 2 .

Edge Coloring of Split Graphs

Abstract A split graph is a graph whose vertex set admits a partition into a stable set and a clique. The chromatic indexes for some subsets of split graphs, such as split graphs with odd maximum degree and split-indifference graphs, are known. However, for the general class, the problem remains unsolved. This paper presents new results about the classification problem for split graphs as a contribution in the direction of solving the entire problem for this class.