Let G be a graph embedded in a surface and let F be a set of even faces of G (faces bounded by a cycle of even length). The resonance graph of G with respect to F, denoted by R(G;F), is a graph such that its vertex set is the set of all perfect matchings of G and two vertices M1 and M2 are adjacent to each other if and only if the symmetric difference M1 ⊕M2 is a cycle bounding some face in F. ...

Let H be a graph. We show that there exists n0 = n0(H) such that for every n ≥ n0, there is a covering of the edges of Kn with copies of H where every edge is covered at most twice and any two copies intersect in at most one edge. Furthermore, the covering we obtain is asymptotically optimal.

We prove that any finite simple graph can be covered by three of its odd subgraphs, and we construct an infinite sequence of graphs where an edge-disjoint covering by three odd subgraphs is not possible. c © ??? John Wiley & Sons, Inc.

We show that any complete k-partite graph G on n vertices, with k ≥ 3, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption that the largest partition class of G contains at most n/2 vertices. This extends known results for complete and complete bipartite

Let H be a graph. We show that there exists n 0 = n 0 (H) such that for every n n 0 , there is a covering of the edges of K n with copies of H where every edge is covered at most twice and any two copies intersect in at most one edge. Furthermore, the covering we obtain is asymptotically optimal.

This paper introduces a scheduling problem with coupledtasks in presence of a compatibility graph on a single processor. We investigate a specific configuration, in which the coupled-tasks possess an idle time equal to 2. The complexity of these problems will be studied according to the presence or absence of triangles in the compatibility graph. As an extended matching, we propose a polynomial...

A circular arc graph is the intersection graph of a collection of connected arcs on the circle. We solve a Turán-type problem for circular arc graphs: for n arcs, if m and M are the minimum and maximum number of arcs that contain a common point, what is the maximum number of edges the circular arc graph can contain? We establish a sharp bound and produce a maximal construction. For a fixed m, t...

The talk will summarize recent works on the stability under disorder of the absolutely continuous spectra of random operators on tree graphs. The cases covered include: Schroedinger operators with random potential, quantum graph operators for trees with randomized edge lengths, and Schroedinger operators with radial quasi-periodic potentials. The results were derived jointly with R. Sims and S....

In this note, we prove that there exists a constant ρ such that any planar graph G of diameter ≤ 2R can be covered with at most ρ balls of radius R, a result conjectured by C. Gavoille, D. Peleg, A. Raspaud, and E. Sopena in 2001.