Mekkia Kouider

Some bounds for the b -chromatic number of a graph

In this paper we study the b-chromatic number of a graph G. This number is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i has at least one representant xi adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. The main result is the determination of two lower bounds for the b-chromatic number of the cartesian product of two graphs.

On the b-dominating coloring of graphs

The b-chromatic number ϕ(G) of a graph G is defined as the largest number k for which the vertices of G can be colored with k colors satisfying the following property: for each i, 1 ≤ i ≤ k, there exists a vertex xi of color i such that for all j ≠ i, 1 ≤ j ≤ k there exists a vertex yj of color j adjacent to xi. A graph G is b-perfect if each induced subgraph H of G has ϕ(H) = χ(H), where χ(H) is the chromatic number of H. We characterize all b-perfect bipartite graphs and all b-perfect P4-sparse graphs by minimal forbidden induced subgraphs. We also prove that every 2K2-free and P-5-free graph is b-perfect.

Bounds for the b-chromatic number of some families of graphs

In this paper we obtain some upper bounds for the b-chromatic number of K1,s-free graphs, graphs with given minimum clique partition and bipartite graphs. These bounds are given in terms of either the clique number or the chromatic number of a graph or the biclique number for a bipartite graph. We show that all the bounds are tight.

Connected Factors in Graphs --- a Survey

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Mean distance and minimum degree

We prove that in a graph of order n and minimum degree d, the mean distance μ must satisfy . This asymptotically confirms, and improves, a conjecture of the computer program GRAFFITI. The result is close to optimal; examples show that for any d, μ may be larger than n/(d + 1).

Edge-vulnerability and mean distance†

The mean distance of a simple connected graph G of order n is defined by n n n n n n nWe answer a problem of Plesnik about the edge-vulnerability of G related to μ, i.e., we find a bound for μ(G − e) − μ(G) and for n n n n n n nwhere e is an edge of G such that G − e is still conencted. This question is of interest in interconnection La distance moyenne dun graphe G, simple et connexe, dordre n, est definie par n n n n n n nNous repondons ici a un probleme de Plesnik sur larěte-vulnerabilite de G par rapport a μ, i.e. nous trouvons une borne pour μ(G −e) − μ(G) et pour n n n n n n nou e est une arěte de G telle que G − e soit encore connexe.

On the existence of a matching orthogonal to a 2-factorization

Abstract This note gives a partial answer to a problem posed by Brian Alspach in a recent issue of Discrete Mathematics. We show that if F1, F2,…,Fd is a 2-factorization of a 2d-regular graph G of order n⩾3.23d then G contains a d-matching with exactly one edge from each of F1, F2,…,Fd.

Connected [a,b]-factors in Graphs

In this paper, we prove the following result:Let G be a connected graph of order n, and minimum degree . Let a and b two integers such that 2a <= b. Suppose and .Then G has a connected [a,b]-factor.

Sufficient Condition for the Existence of an Even [a, b]-Factor in Graph

Let a, b, be two even integers. In this paper, we get a sufficient condition which involves the stability number, the minimum degree of the graph for the existence of an even [a, b]-factor.

Decomposition of multigraphs

In this note, we consider the problem of existence of an edgedecomposition of a multigraph into isomorphic copies of 2-edge paths K1,2. We find necessary and sufficient conditions for such a decomposition of a multigraph H to exist when (i) either H does not have incident multiple edges or (ii) multiplicities of the edges in H are not greater than two. In particular, we answer a problem stated by Z. Skupień.