42 research outputs found
Clique number of tournaments
We introduce the notion of clique number of a tournament and investigate its
relation with the dichromatic number. In particular, it permits defining
\dic-bounded classes of tournaments, which is the paper's main topic
Convex circuit free coloration of an oriented graph
We introduce the \textit{convex circuit-free coloration} and \textit{convex circuit-free chromatic number} of an oriented graph and establish various basic results. We show that the problem of deciding if an oriented graph verifies is NP-complete if and polynomial if . We exhibit an algorithm which finds the optimal convex circuit-free coloration for tournaments, and characterize the tournaments that are \textit{vertex-critical} for the convex circuit-free coloration
The 3-dicritical semi-complete digraphs
A digraph is -dicritical if it cannot be vertex-partitioned into two sets
inducing acyclic digraphs, but each of its proper subdigraphs can. We give a
human-readable proof that the number of 3-dicritical semi-complete digraphs is
finite. Further, we give a computer-assisted proof of a full characterization
of 3-dicritical semi-complete digraphs. There are eight such digraphs, two of
which are tournaments. We finally give a general upper bound on the maximum
number of arcs in a -dicritical digraph
Uniquely colourable m-dichromatic oriented graphs
AbstractThe dichromatic number dk(D) of a diagraph D is the minimum number of colours needed to colour V(D) in such a way that no monochromatic directed cycle is obtained. A digraph D is called uniquely colourable if any acyclic dk(D)-colouring of V(D) induces the same partition of V(D). In this paper we construct an infinite family of uniquely colourable m-dichromatic oriented graphs for all m ⩾ 2
Sharp bounds for Laplacian spectral moments of digraphs with a fixed dichromatic number
The k-th Laplacian spectral moment of a digraph G is defined as ∑i=1nλik, where λi are the eigenvalues of the Laplacian matrix of G and k is a nonnegative integer. For k=2, this invariant is better known as the Laplacian energy of G. We extend recently published results by characterizing the digraphs which attain the minimal and maximal Laplacian energy within classes of digraphs with a fixed dichromatic number. We also determine sharp bounds for the third Laplacian spectral moment within the special subclass which we define as join digraphs. We leave the full characterization of the extremal digraphs for k≥3 as an open problem.</p
The smallest 5-chromatic tournament
A coloring of a digraph is a partition of its vertex set such that each class
induces a digraph with no directed cycles. A digraph is -chromatic if is
the minimum number of classes in such partition, and a digraph is oriented if
there is at most one arc between each pair of vertices. Clearly, the smallest
-chromatic digraph is the complete digraph on vertices, but determining
the order of the smallest -chromatic oriented graphs is a challenging
problem. It is known that the smallest -, - and -chromatic oriented
graphs have , and vertices, respectively. In 1994, Neumann-Lara
conjectured that a smallest -chromatic oriented graph has vertices. We
solve this conjecture and show that the correct order is
The A<sub>α</sub> spectral moments of digraphs with a given dichromatic number
The Aα-matrix of a digraph G is defined as Aα(G)=αD+(G)+(1−α)A(G), where α∈[0,1), D+(G) is the diagonal outdegree matrix and A(G) is the adjacency matrix. The k-th Aα spectral moment of G is defined as ∑i=1 nλαi k, where λαi are the eigenvalues of the Aα-matrix of G, and k is a nonnegative integer. In this paper, we obtain the digraphs which attain the minimal and maximal second Aα spectral moment (also known as the Aα energy) within classes of digraphs with a given dichromatic number. We also determine sharp bounds for the third Aα spectral moment within the special subclass which we define as join digraphs. These results are related to earlier results about the second and third Laplacian spectral moments of digraphs.</p