Using the idea due to P. DUCHET in proving his well–known theorem on powers of chordal graphs, we shall describe some theorems of DUCHET–type for powers of graphs that have no long induced cycles. In particular, our DUCHET–type theorem for HHD–free graphs improves a recent result due to DRAGAN, NICOLAI, BRANDSTÄDT saying that odd powers of HHD–free graphs are also HHD–free.

Let k be an integer and k ≥ 3. A graph G is k-chordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3-chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if G is chordal then so is G. Brandstädt et al. in [Andreas Brandstädt, Van Bang Le, and Thomas Szymczak. Duchet-type theorems for pow...

The Conjecture of Hadwiger implies that the Hadwiger number h times the independence number α of a graph is at least the number of vertices n of the graph. In 1982 Duchet and Meyniel proved a weak version of the inequality, replacing the independence number α by 2α− 1, that is, (2α− 1) · h ≥ n. In 2005 Kawarabayashi, Plummer and the second author published an improvement of the theorem, replaci...

In this paper we prove that perfect graphs are kernel solvable, as it was conjectured by Berge and Duchet (1983). The converse statement, i.e. that kernel solvable graphs are perfect, was also conjectured in the same paper, and is still open. In this direction we prove that it is always possible to substitute some of the vertices of a non-perfect graph by cliques so that the resulting graph is ...

Probably the best-known among the remaining unsolved problems in graph theory is Hadwiger’s Conjecture: If χ(G) denotes the chromatic number of graph G, then G has the complete graph Kχ(G) as a minor. The following would immediately follow from the truth of Hadwiger’s Conjecture: Conjecture: If G has n vertices and if α(G) denotes the independence number of G, then G has Kdn/α(G)e as a minor. I...

Since χ(G) · α(G) ≥ |V (G)|, Hadwiger’s Conjecture implies that any graph G has the complete graph Kdn α e as a minor, where n is the number of vertices of G and α is the maximum number of independent vertices in G. Motivated by this fact, it is shown that any graph on n vertices with independence number α ≥ 3 has the complete graph Kd n 2α−2 e as a minor. This improves the well-known theorem o...