Victor A. Campos

On the asymmetric representatives formulation for the vertex coloring problem

We consider the vertex coloring problem, which can be stated as the problem of minimizing the number of labels that can be assigned to the vertices of a graph G such that each vertex receives at least one label and the endpoints of every edge are assigned different labels. In this work, the 0-1 integer programming formulation based on representative vertices is revisited to remove symmetry. The previous polyhedral study related to the original formulation is adapted and generalized. New versions of facets derived from substructures of G are presented, including cliques, odd holes and anti-holes and wheels. In addition, a new class of facets is derived from independent sets of G. Finally, a comparison with the independent sets formulation is provided.

b-chromatic number of cacti☆

Abstract A b-colouring of a graph G is a proper colouring of G such that each colour contains a vertex that is adjacent to all other colours and the b-chromatic number χ b ( G ) is the maximum number of colours used in a b-colouring of G. If m ( G ) is the largest integer k such that G has at least k vertices with degree at least k − 1 , then we know that χ b ( G ) ⩽ m ( G ) . Irving and Manlove [Irving, R.W. and Manlove, D.F., The b-chromatic number of a graph, Discrete Applied Mathematics, 91 (1999), pages 127–141] prove that, if T is a tree, then the b-chromatic number of T is at least m ( T ) − 1 . In this paper, we prove that, if G is a connected cactus and m ( G ) ⩾ 7 , then the b-chromatic number of G is at least m ( G ) − 1 .

The maximum time of 2-neighbour bootstrap percolation

In 2-neighbourhood bootstrap percolation on a graph G , an infection spreads according to the following deterministic rule: infected vertices of G remain infected forever and in consecutive rounds healthy vertices with at least 2 already infected neighbours become infected. Percolation occurs if eventually every vertex is infected. In this paper, we are interested to calculate the maximal time t ( G ) the process can take, in terms of the number of times the interval function is applied, to eventually infect the entire vertex set. We prove that the problem of deciding if t ( G ) ? k is NP-complete for: (a) fixed k ? 4 ; (b) bipartite graphs and fixed k ? 7 ; and (c) planar graphs. Moreover, we obtain linear and polynomial time algorithms for trees and chordal graphs, respectively.

Graphs with few P 4 's under the convexity of paths of order three

A graph is ( q , q - 4 ) if every subset of at most q vertices induces at most q - 4 P 4 s. It therefore generalizes some different classes, as cographs and P 4 -sparse graphs. In this work, we propose algorithms for determining various NP-Hard graph convexity parameters within the convexity of paths of order three, for ( q , q - 4 ) graphs. All algorithms have linear-time complexity, for fixed q , and then are fixed parameter tractable. Moreover, we prove that the Caratheodory number is at most three for every cograph, P 4 -sparse graph and every connected ( q , q - 4 ) -graph with at least q vertices.

b-Coloring graphs with large girth

A b-coloring of a graph is a coloring of its vertices such that every color class contains a vertex that has a neighbor in all other classes. The b-chromatic number of a graph is the largest integer k such that the graph has a b-coloring with k colors. We show how to compute in polynomial time the b-chromatic number of a graph of girth at least 9. This improves the seminal result of Irving and Manlove on trees.

The b-chromatic index of graphs

A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to at least one vertex in each other color class. The b-chromatic number of G is the maximum integer b ( G ) for which G has a b-coloring with b ( G ) colors. This problem was introduced by Irving and Manlove (1999), where they showed that computing b ( G ) is NP -hard in general and polynomial-time solvable for trees. A natural question that arises is whether the edge version of this problem is also NP -hard or not. Here, we prove that computing the b-chromatic index of a graph G is NP -hard, even if G is either a comparability graph or a C k -free graph, and give partial results on the complexity of the problem restricted to trees, more specifically, we solve the problem for caterpillars graphs. Although solving problems on caterpillar graphs is usually quite simple, this problem revealed itself to be unusually hard. The presented algorithm uses a dynamic programming approach that combines partial solutions which are proved to exist if, and only if, a particular polyhedron is non-empty.

Graphs of girth at least 7 have high b-chromatic number

A b-coloring of a graph is a proper coloring of its vertices such that every color class contains a vertex that has neighbors in all other color classes. The b-chromatic number of a graph is the largest integer b ( G ) such that the graph has a b-coloring with b ( G ) colors. This metric is upper bounded by the largest integer m ( G ) for which G has at least m ( G ) vertices with degree at least m ( G ) - 1 . There are a number of results reporting that graphs with high girth have high b-chromatic number when compared to m ( G ) . Here, we prove that every graph with girth at least 7 has b-chromatic number at least m ( G ) - 1 . Our proof also yields a polynomial algorithm that produces an optimal b-coloring of these graphs.

Transversals in Trees

A transversal in a rooted tree is any set of nodes that meets every path from the root to a leaf. We let c(T,k) denote the number of transversals of size k in a rooted tree T. We define a partial order on the set of all rooted trees with n nodes by saying that a tree T succeeds a tree T′ if c(T,k) is at least c(T′,k) for all k and strictly greater than c(T′,k) for at least one k. We prove that, for every choice of positive integers d and n, the set of all rooted trees on n nodes where each node has at most d children has a unique minimal element with respect to this partial order and we describe this tree. In addition, we determine asymptotically the expected values of c(T,k) in special families of trees. C

New Bounds on the Grundy Number of Products of Graphs

The Grundy number of a graph G is the largest k such that G has a greedy k-coloring, that is, a coloring with k colors obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this article, we give new bounds on the Grundy number of the product of two graphs.

Restricted coloring problems on graphs with few P4′s

Abstract In this paper, we obtain polynomial time algorithms to determine the acyclic chromatic number, the star chromatic number and the harmonious chromatic number of P 4 -tidy graphs and ( q , q − 4 )-graphs, for every fixed q. These classes include cographs, P 4 -sparse and P 4 -lite graphs. We also obtain a polynomial time algorithm to determine the Grundy number of ( q , q − 4 )-graphs. All these coloring problems are known to be NP-hard for general graphs.