Odile Favaron

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.

Contributions to the theory of domination, independence and irredundance in graphs

A vertex x in a subset X of vertices of an undirected graph is redundant if its closed neighbourhood is contained in the union of closed neighbourhoods of vertices of X-{x}. In the context of a communications network, this means that any vertex which may receive communications from X may also be informed from X-{x}. The lower and upper irredundance numbers ir(G) and IR(G) are respectively the minimum and maximum cardinalities taken over all maximal sets of vertices having no redundancies. The domination number @c(G) and upper domination number @C(G) are respectively the minimum and maximum cardinalities taken over all minimal dominating sets of G. The independent domination number i(G) and the independence number @b(G) are respectively the minimum and maximum cardinalities taken over all maximal independent sets of vertices of G. A variety of inequalities involving these quantities are established and sufficient conditions for the equality of the three upper parameters are given. In particular a conjecture of Hoyler and Cockayne [9], namely i+@b=<2p + 2@d - 22p@d, is proved.

Signed domination in regular graphs

Abstract In answer to the open questions proposed by Henning and Slater, we give sharp upper bounds on the upper signed domination number of a regular graph and on the signed domination number of a connected cubic graph.

Paired-Domination in Claw-Free Cubic Graphs

Abstract.A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The minimum cardinality of a paired-dominating set of G is the paired-domination number of G, denoted by γpr(G). If G does not contain a graph F as an induced subgraph, then G is said to be F-free. In particular if F=K1,3 or K4−e, then we say that G is claw-free or diamond-free, respectively. Let G be a connected cubic graph of order n. We show that (i) if G is (K1,3,K4−e,C4)-free, then γpr(G)≤3n/8; (ii) if G is claw-free and diamond-free, then γpr(G)≤2n/5; (iii) if G is claw-free, then γpr(G)≤n/2. In all three cases, the extremal graphs are characterized.

Total domination in graphs with minimum degree three

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph that is reducible by some finite sequence of these moves, to a graph with no edges, is called a knot graph. We show that the class of knot graphs strictly contains the set of delta-wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes.

Offensive alliances in graphs

A set S is an offensive alliance if for every vertex v in its boundary N(S)−S it holds that the majority of vertices in v’s closed neighbourhood are in S. The offensive alliance number is the minimum cardinality of an offensive alliance. In this paper we explore the bounds on the offensive alliance and the strong offensive alliance numbers (where a strict majority is required). In particular, we show that the offensive alliance number is at most 2/3 the order and the strong offensive alliance number is at most 5/6 the order.

On the Randić index

The Randic index R(G) of a graph G = (V, E) is the sum of (d(u)d(υ))-1/2 over all edges uυ ∈ E of G. Bollobas and Erdos (Ars Combin. 50 (1998) 225) proved that the Randic index of a graph of order n without isolated vertices is at least √n - 1. They asked for the minimum value of R(G) for graphs G with given minimum degree δ(G). We answer their question for δ(G) = 2 and propose a related conjecture. Furthermore, we prove a best-possible lower bound on the Randic index of a triangle-free graph G with given minimum degree δ(G).

On k-factor-critical graphs

A graph is said to be k-factor-critical if the removal of any set of k vertices results in a graph with a perfect matching. We study some properties of k-factor-critical graphs and show that many results on q-extendable graphs can be improved using this concept.

Closure and stable Hamiltonian properties in claw-free graphs

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph that is reducible by some finite sequence of these moves, to a graph with no edges, is called a knot graph. We show that the class of knot graphs strictly contains the set of delta-wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes.