Cláudia Linhares Sales

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 .

On minimally b-imperfect graphs

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbour in all other color classes. The b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph H of G. A graph is minimally b-imperfect if it is not b-perfect and every proper induced subgraph is b-perfect. We give a list F of minimally b-imperfect graphs, conjecture that a graph is b-perfect if and only if it does not contain a graph from this list as an induced subgraph, and prove this conjecture for diamond-free graphs, and graphs with chromatic number at most three.

Oriented trees in digraphs

Abstract Let f ( k ) be the smallest integer such that every f ( k ) -chromatic digraph contains every oriented tree of order k . Burr proved f ( k ) ≤ ( k − 1 ) 2 in general, and he conjectured f ( k ) = 2 k − 2 . Burr also proved that every ( 8 k − 7 ) -chromatic digraph contains every antidirected tree. We improve both of Burr’s bounds. We show that f ( k ) ≤ k 2 / 2 − k / 2 + 1 and that every antidirected tree of order k is contained in every ( 5 k − 9 ) -chromatic digraph. We make a conjecture that explains why antidirected trees are easier to handle. It states that if | E ( D ) | > ( k − 2 ) | V ( D ) | , then the digraph D contains every antidirected tree of order k . This is a common strengthening of both Burr’s conjecture for antidirected trees and the celebrated Erdős-Sos Conjecture. The analogue of our conjecture for general trees is false, no matter what function f ( k ) is used in place of k − 2 . We prove our conjecture for antidirected trees of diameter 3 and present some other evidence for it. Along the way, we show that every acyclic k -chromatic digraph contains every oriented tree of order k and suggest a number of approaches for making further progress on Burr’s conjecture.

Even Pairs in Claw-Free Perfect Graphs

An even pair in a graph is a pair of non-adjacent vertices such that every chordless path between them has even length. A graph is called strict quasi-parity when every induced subgraph that is not a clique has an even pair, and it is called perfectly contractile when every induced subgraph can be turned into a clique through a sequence of even-pair contractions. In this paper we determine theK1,3-free graphs that are strict quasi-parity and those that are perfectly contractile. We show that for both classes the minimal forbidden configurations are odd holes, antiholes and some line-graphs of bipartite graphs, as conjectured by several authors. Our proofs are constructive and yield polynomial-time algorithms for the recognition of both classes.

On Planar Perfectly Contractile Graphs

An even pair in a graph is a pair of vertices such that every chordless path between them has even length. A graph is called perfectly contractile when every induced subgraph can be transformed into a clique through a sequence of even-pair contractions. In this paper we characterize the planar graphs that are perfectly contractile by determining all the minimal forbidden subgraphs. We give a polynomial algorithm for the recognition of perfectly contractile planar graphs.

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.

Grundy number on P4-classes

Abstract In this article, we define a new class of graphs, the fat-extended P 4 -laden graphs, and we show a polynomial time algorithm to determine the Grundy number of the graphs in this class. This result implies that the Grundy number can be found in polynomial time for any graph of the following classes: P4-reducible, extended P 4 -reducible, P 4 -sparse, extended P 4 -sparse, P 4 -extendible, P 4 -lite, P 4 -tidy, P 4 -laden and extended P 4 -laden, which are all strictly contained in the fat-extended P 4 -laden class.

b-coloring of m-tight graphs☆

Abstract A given k-coloring c of a graph G = ( V , E ) is a b-coloring if for every color class c i , 1 ⩽ i ⩽ k , there is a vertex colored i whose neighborhood intersect every other color class c j of c. The b-chromatic number of G is the greatest integer k such that G admits a b-coloring with k colors. A graph G is m-tight if it has exactly m = m ( G ) vertices of degree exactly m − 1 , where m ( G ) is the largest integer m such that G has at least m vertices of degree at least m − 1 . Determining the b-chromatic number of a m-tight graph G is NP-hard [Jan Kratochvil, Zsolt Tuza, and Margit Voigt. On the b-chromatic number of graphs. In Proc. of the 28th International Workshop on Graph-Theoretic Concepts in Computer Science, pages 310–320. Springer-Verlag, 2002]. In this paper, we define the b-closure and the partial b-closure of a m-tight graph. These concepts were used to give a characterization of m-tight graphs whose b-chromatic number is equal to m. To illustrate an application of our characterization, we provide some examples for which we can answer in polynomial time whether χ b ( G ) m . We also generalized the definition of pivoted tree introduced by Irving and Manlove [Robert W. Irving and David F. Manlove. The b-chromatic number of a graph. Discrete Appl. Math., 91(1-3):127–141, 1999] and show how the existence of our pivots affect the behavior of the b-chromatic number parameter in m-tight graphs.

b-continuity and the lexicographic product of graphs☆

Abstract A k-coloring c of a graph G = ( V , E ) is a b-coloring if for every color class c i , 1 ≤ i ≤ k , there is a vertex colored i whose neighborhood intersects every other color class c j of c. The b-chromatic number of G , χ b ( G ) , is the greatest k such that G admits a b-coloring with k colors. Every optimal coloring of G is a b-coloring. Therefore χ ( G ) ≤ χ b ( G ) . G is b-continuous if for every k , χ ( G ) ≤ k ≤ χ b ( G ) , G admits a b-coloring with k colors. In this paper, we are interested in b-continuous graphs G [ H ] which are the lexicographic product of two b-continuous graphs G and H. We give partial results on the spectrum of G [ H ] and we examine its b-continuity for specific classes of G and H.

On the b-Continuity of the Lexicographic Product 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 each other color class. The b-chromatic number of G is the maximum integer