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Given a graph G, its partially square graph G* is a graph obtained by adding an edge (u,v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition $N_G\left(x\right)\subseteq N_G\left[u\right]\cup N_G\left[v\right]$, where $N_G\left[x\right]=N_G\left(x\right)\cup x$. In the case where G is a claw-free graph, G* is equal to G². We define $\sigma °ₜ=min{\sum }_{x\in S}{d}_G\left(x\right):SisanindependentsetinG*and\left|S\right|=t$. We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results.

The Grundy number of a graph G is the maximum number k of colors used to color the vertices of G such that the coloring is proper and every vertex x colored with color i, 1 ≤ i ≤ k, is adjacent to (i-1) vertices colored with each color j, 1 ≤ j ≤ i -1. In this paper we give bounds for the Grundy number of some graphs and cartesian products of graphs. In particular, we determine an exact value of this parameter for n-dimensional meshes and some n-dimensional toroidal meshes. Finally, we present an...

H. Kheddouci, J.F. Saclé and M. Woźniak conjectured in 2000 that if a tree T is not a star, then there is an edge-disjoint placement of T into its third power.In this paper, we prove the conjecture for caterpillars.

Let k be a positive integer, Sk and Ck denote, respectively, a star and a cycle of k edges. λKn is the usual notation for the complete multigraph on n vertices and in which every edge is taken λ times. In this paper, we investigate necessary and sufficient conditions for the existence of the decomposition of λKn into edges disjoint of stars Sk’s and cycles Ck’s.