Victor Neumann-Lara

The dichromatic number of a digraph

Abstract In this paper the concept of dichromatic number of a digraph which is a generalization of the chromatic number of a graph is introduced. The dichromatic number of a digraph D is defined as the minimum number of colours required to colour the vertices of D in such a way that the chromatic classes induce acyclic subdigraphs in D . Some results relating the dichromatic number of D with the existence of cycles of special lengths in D are presented. Contributions to chromatic theory are also obtained. In particular, we generalize the theorem due to P. Erdos and A. Hajnal ( Acta Math. Acad. Sci. Hungar. 17 (1966), 61–99) which states the existence of odd cycles of length ≥ χ ( G ) − 1 in any graph G .

On kernels and semikernels of digraphs

A kernel N of a digraph D is an independent set of vertices of D such that for every w @? V(D) - N there exists an arc from w to N. If every induced subdigraph of D has a kernel, D is said to be an R-digraph. Minimal non-R-digraphs are called R^--digraphs. In this paper some structural results concerning R^--digraphs and sufficient conditions for a digraph to be an R-digraph are presented. In particular, it is proved that every vertex (resp. arc) in an R^--digraph is contained in an odd directed cycle not containing special pseudodiagonals. It is also proved that any digraph in which every odd directed cycle has two pseudodiagonals with consecutive terminal endpoints is an R-digraph. Previous results of other authors (Richardson, Meyniel, Duchet, and others) are generalized.

On the minimum size of tight hypergraphs

A k-graph, H = (V, E), is tight if for every surjective mapping f: V {1,….k} there exists an edge α ϵ E sicj tjat f|α is injective. Clearly, 2-graphs are tight if and only if they are connected. Bounds for the minimum number ϕ of edges in a tight k-graph with n vertices are given. We conjecture that ϕ for every n and prove the equality when 2n + 1 is prime. From the examples, minimal embeddings of complete graphs into surfaces follow.

On kernel-perfect critical digraphs

In this paper we investigate new sufficient conditions for a digraph to be kernel-perfect (KP) and some structural properties of kernel-perfect critical (KPC) digraphs. In particular, it is proved that the asymmetrical part of any KPC digraph is strongly connected. A new method to construct KPC digraphs is developed. The existence of KP and KPC digraphs with arbitrarily large dichromatic number is also discussed.

An Anti-Ramsey Theorem

Let be the Turán number which gives the maximum size of a graph of order containing no subgraph isomorphic to .In 1973, Erdős, Simonovits and Sós [5] proved the existence of an integer such that for every integer , the minimum number of colours , such that every -colouring of the edges of which uses all the colours produces at least one all whose edges have different colours, is given by . However, no estimation of was given in [5]. In this paper we prove that for . This formula covers all the relevant values of n and p.

An Anti-Ramsey Theorem on Cycles

Let h(n,p) be the minimum integer such that every edge-colouring of the complete graph of order n, using exactly h(n,p) colours, produces at least one cycle of order p having all its edges of different colours. In this paper the value of h(n,p) is determinated for n≥p≥3. As a corollary we obtain the equality which was conjectured by Erdös, Simonovits and Sós, 30 years ago [4].

Locally C 6 graphs are clique divergent

The clique graph kG of a graph G is the intersection graph of the family of all maximal complete subgraphs of G. The iterated clique graphs knG are defined by k0G=G and kn+1G=kknG. A graph G is said to be k-divergent if V(knG) tends to infinity with n. A graph is locally C6 if the neighbours of any given vertex induce an hexagon. We prove that all locally C6 graphs are k-divergent and that the diameters of the iterated clique graphs also tend to infinity with n while the sizes of the cliques remain bounded.

Whitney triangulations, local girth and iterated clique graphs

We study the dynamical behaviour of surface triangulations under the iterated application of the clique graph operator k, which transforms each graph G into the intersection graph kG of its (maximal) cliques. A graph G is said to be k-divergent if the sequence of the orders of its iterated clique graphs |V (k n G)| tends to in4nity with n. If this is not the case, then G is eventually k-periodic, or k-bounded: k n G ∼ k m G for some m?n . The case in which G is the underlying graph of a regular triangulation of some closed surface has been previously studied under the additional (Whitney) hypothesis that every triangle of G is a face of the triangulation: if G is regular of degree d, it is known that G is k-bounded for d = 3 and k-divergent for d = 4; 5; 6. We will show that G is k-bounded for all d ? 7, thus completing the study of the regular case. Our proof works in the more general setting of graphs with local girth at least 7. As a consequence we obtain also the k-boundedness of the underlying graph G of any triangulation of a compact surface (with or without border) provided that any triangle of G is a face of the triangulation and that the minimum degree of the interior vertices of G is at least 7. c � 2002 Published by Elsevier Science B.V.

A Family of Clique Divergent Graphs with Linear Growth

We present an infinite set A of finite graphs such that for any graph G e A the order | V(kn(G))| of the n-th iterated clique graph kn(G) is a linear function of n. We also give examples of graphs G such that | V(kn(G))| is a polynomial of any given positive degree.

Clique divergent graphs with unbounded sequence of diameters

Abstract The clique graph kG of a graph G is the intersection graph of the family of all maximal complete subgraphs of G . The iterated clique graphs k n G are defined by k 0 G = G and k n +1 G = kk n G . A graph G is said to be k -divergent if | V ( k n G )| tends to infinity with n . We provide examples of k -divergent graphs such that the diameters of the iterated clique graphs also tend to infinity with n . Furthermore, the sizes of the cliques and even the chromatic numbers remain bounded.