Maryvonne Mahéo

Hamiltonian decomposition of Cayley graphs of degree 4

Abstract We prove that any 4-regular connected Cayley graph on a finite abelian group can be decomposed into two hamiltonian cycles. This answers a partial case of Alspachs conjecture concerning hamiltonian decompositions of 2k -regular connected Cayley graphs. As a corollary we obtain the hamiltonian decomposition of 2-jump circulant graphs, also called double loops.

Some eigenvalue properties in graphs (conjectures of Graffiti. II)

In this paper we improve some classical bounds on the greatest eigenvalue of the adjuacency matrix of a graph. We also give inequalities between the eigenvalues and some other parameters. These results allow us to prove some conjectures of the program Graffiti written by Fajtlowicz. Moreover, the study of the spectrum of graphs obtained by some simple constructions yields infinite families of counterexamples for other conjectures of this program.

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 residue of a graph

The residue R of a simple graph G of degree sequence S:d 1 ≥d 2 ≥...≥d n is the number of zeros obtained by the iterative process consisting of deleting the first term d 1 of S, subtracting 1 from the d 1 following ones, and sorting down the new sequence. The depth is the number n−R of steps in this algorithm. We prove here some conjectures given by the computer program GRAFFITI, in particular: R≤α, where α is the independence number of G. If G is a tree: μ ≤R, where μ is the mean distance of G

Some minimum broadcast graphs

Abstract In this paper, we give the size B(n) of a minimum broadcast graph of order n for the values 20, 21 and 22 of n , improve the upper bound of B (23), and by corollary of some other B(n) .

Isomorphisms of Cayley multigraphs of degree 4 on finite Abelian groups

Abstract The correspondence between finite abelian groups and their Cayley graphs is studied in the case of degree 4. We show that, but for a simple family of exceptions, the graph is sufficient to determine up to isomorphism the group and the set giving the edges. Ada´ms conjecture about isomorphisms of circulant graphs with degree 4 is thus proven.

Edge-disjoint placement of three trees

Abstract We present complete results concerning edge-disjoint placement of three trees of orderninto the complete graphkn.

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.

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.

Note on the problem of gossiping in multidimensional grids

Abstract Recently, we solved a problem which we found in a list given at the international workshop on Broadcasting and Gossiping in Vancouver (July 1990). This problem asked for the minimum time required to gossip in multidimensional nontoroidal grids. However, we found later that this question, asked by Krumme, had already been solved ten years ago by Farley and Proskurowski (1980) in the case of two-dimensional grids. In their paper they did not mention anything about higher dimensions, although the result can be easily deduced as we explain in this paper.