On clique‐inverse graphs of graphs with bounded clique number

The clique graph K(G) of G is the intersection graph of the family of maximal cliques of G. For a family F of graphs, the family of clique-inverse graphs of F, denoted by K−1(F), is defined as K−1(F) = {H|K(H) ∈ F}. Let F p be the family of Kp-free graphs, that is, graphs with clique number at most p − 1, for an integer constant p ≥ 2. Deciding whether a graph H is a clique-inverse graph of F p can be done in polynomial time; in addition, for p ∈ {2, 3, 4}, K − 1 (Fp) can be characterized by a finite family of forbidden induced subgraphs. In Protti and Szwarcfiter, the authors propose to extend such characterizations to higher values of p. Then a natural question arises: Is there a characterization of K − 1 (Fp) by means of a finite family of forbidden induced subgraphs, for any p ≥ 2? In this note we give a positive answer to this question. We present upper bounds for the order, the clique number, and the stability number of every forbidden induced subgraph for K − 1 (Fp) in terms of p.Facultad de Ciencias Exacta

The Proportional Colouring Problem: Optimizing Buffers in Wireless Mesh Networks

In this paper, we consider a new edge colouring problem motivated by wireless mesh networks optimization: the proportional edge colouring problem. Given a graph G with positive weights associated to its edges, we want to find a proper edge colouring which assigns to each edge at least a proportion (given by its weight) of all the colours. If such colouring exists, we want to find one using the minimum number of colours. We proved that deciding if a weighted graph admits a proportional edge colouring is polynomial while determining its proportional edge chromatic number is NP-hard. We also give a lower and an upper bound that can be polynomially computed. We finally characterize some graphs and weighted graphs for which we can determine the proportional edge chromatic number

b-coloring of tight graphs

A given k-coloring c of a graph G = (V,E) is a b-coloring if for every color class ci, 1 ≤ i ≤ k, there is a vertex colored i whose neighborhood intersects every other color class cj 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 tight if it has exactly m(G) vertices of degree exactly m(G)−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 tight graph G is NP-hard even for a connected bipartite graph [9]. In this pa- per we show that it is also NP-hard for a tight chordal graph. We also consider the particular case where the input graph is split (not necessarily tight) and show that the problem of determining the b-chromatic number is polynomial in this case. Then we define the b-closure and the partial b-closure of a tight graph, and use these concepts to give a characteriza- tion of tight graphs whose b-chromatic number is equal to m(G). This characterization is used to develop polynomial time algorithms for decid- ing whether χb(G) < m, for tight graphs that are complement of bipartite graphs, P4-sparse and block graphs. We generalize the concept of pivoted tree introduced by Irving and Manlove [6] and show its relation with the behavior of the b-chromatic number of tight graphs