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Remarks on partially square graphs, hamiltonicity and circumference

100%
Kheddouci H.
Discussiones Mathematicae Graph Theory
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2001
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tom 21
|
nr 2
255-266
EN
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 $N_G(x) ⊆ N_G[u] ∪ N_G[v]$, where $N_G[x] = N_G(x) ∪ {x}$. In the case where G is a claw-free graph, G* is equal to G². We define $σ°ₜ = min{ ∑_{x∈S} d_G(x):S is an independent set in G* and |S| = t}$. We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results.
2
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On the packing of two copies of a caterpillar in its third power

63%
Germain C., Kheddouci H.
Discussiones Mathematicae Graph Theory
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2003
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tom 23
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nr 1
105-115
EN
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.
3
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Grundy number of graphs

63%
Effantin B., Kheddouci H.
Discussiones Mathematicae Graph Theory
|
2007
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tom 27
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nr 1
5-18
EN
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 algorithm to generate all graphs for a given Grundy number.
4
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Decomposition of Complete Multigraphs Into Stars and Cycles

51%
Beggas F., Haddad M., Kheddouci H.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
|
nr 4
629-639
EN
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.
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