Cycles in partially square graphs

In this work we consider finite undirected simple graphs. If C = (V, E) is a graph we denote by alpha (G) the stability number of G. For any vertex v let N[x] be the union of x and the neighborhood N(x). For each pair of vertices a b of G we associate the set J(a, b) as follows. J(a, b) = {u is an element of N[a] boolean AND N[b] \ N(u) subset of or equal to N[a] boolean OR N[b]). Given a graph G. its partially square G* is the graph obtained by adding an edge uv for each pair It, v of vertices of G at distance 2 whenever J(u, 2,) is not empty. In the case G is a claw-free graph, G* is equal to If G is k-connected, we cover the vertices of G by at most [alpha (G*)/k] cycles, where alpha (G*) is the stability number of the partially square graph of G. On the other hand we consider in G* conditions on the sum of the degrees. Let G be any 2-connected graph and t be any integer (t greater than or equal to 2). If Sigma (x is an element ofS) deg(G)(x) greater than or equal to1 \G\, for every t-stable set S subset of or equal to V(G) of G* then the vertex set of G can be covered with t-1 cycles. Different corollaries on covering by paths are given.