On the family of $r$-regular graphs with Grundy number $r+1$

International audience; The Grundy number of a graph $G$, denoted by $\Gamma(G)$, is the largest $k$ such that there exists a partition of $V(G)$, into $k$ independent sets $V_1,\ldots, V_k$ and every vertex of $V_i$ is adjacent to at least one vertex in $V_j$, for every $j < i$. The objects which are studied in this article are families of $r$-regular graphs such that $\Gamma(G) = r + 1$. Using the notion of independent module, a characterization of this family is given for $r=3$. Moreover, we determine classes of graphs in this family, in particular the class of $r$-regular graphs without induced $C_4$, for $r \le 4$. Furthermore, our propositions imply results on partial Grundy number.

Partially Square Graphs, Hamiltonicity and Circumference II

Abstract Given a graph G, its partially square graph G∗ is a graph obtained by adding an edge uv 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 NG(x) ⊆ NG[u] ∪ NG[v], where NG[x]= NG(x) ∪ {x}. In case G is a claw-free graph, G∗ is equal to G2, We define σ ∗ t = min{ ∑ x∈ d ∗ G (x): S is an independent set in G ∗ and ∣S∣ = t} , where d ∗ G (x) = ∣{y ∈ V∣ xy ∈ E(G∗)}∣ . We give for hamiltonicity and circumference new sufficient conditions depending on and we improve some known results.

Adjacent vertex distinguishing edge-colorings of meshes and hypercubes

International audience

A note on the packing of two copies of some trees into their third power

Abstract It is proved in [1] that if a tree T of order n is not a star, then there exists an edge-disjoint placement of two copies of this tree into its fourth power. In this paper, we prove the packing of some trees into their third power.

Grundy coloring for power graphs

International audience

A dual of 4-regular graph forG × C2n

Abstract A graph is said h-decomposable if its edge-set is decomposable into edge-disjoint hamiltonian cycles. Jha [3] conjectured that if G is a non-bipartite h-decomposable graph on even number of vertices, then G × K2 is h-decomposable. We use the notion of dual graph defined in [4], we prove that if G = Q1,2 ⊕ C3,4 is a 4-regular non-bipartite h-decomposable graph and the dual graphs relative to Q1,2 and C3,4 are connected then G × K 2 and G × C 2n are h-decomposable (where C 2n is an even cycle).

The b-chromatic number of power graphs

The b-chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i≤ k, has at least one representant x_i adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. In this paper, we discuss the b-chromatic number of some power graphs. We give the exact value of the b-chromatic number of power paths and power complete binary trees, and we bound the b-chromatic number of power cycles.

Packing of two copies of a caterpillar into its third power

International audience

The irregularity strength of circulant graphs

AbstractThe irregularity strength of a simple graph is the smallest integer k for which there exists a weighting of the edges with positive integers at most k such that all the weighted degrees of the vertices are distinct. In this paper we study the irregularity strength of circulant graphs of degree 4. We find the exact value of the strength for a large family of circulant graphs.

Skeletizing 3D-Objects by Projections

Skeletization is used to simplify an object and to give an idea of the global shape of an object. This paper concerns the continuous domain. While many methods already exist, they are mostly applied in 2D-space. We present a new method to skeletize the polygonal approximation of a 3D-object, based on projections and 2D-skeletization from binary trees.

Subdivision into i-packings and S-packing chromatic number of some lattices

An ?$i$?-packing in a graph ?$G$? is a set of vertices at pairwise distance greater than ?$i$?. For a nondecreasing sequence of integers ?$S=(s_1,s_2,\ldots)$?, the?$S$?-packing chromatic number of a graph ?$G$? is the least integer ?$k$? such that there exists a coloring of ?$G$? into ?$k$? colors where each set of vertices colored ?$i$?, ?$i=1,\ldots,k$?, is an ?$s_i$?-packing. This paper describes various subdivisions of an ?$i$?-packing into ?$j$?-packings ?$(j>i)$? for the hexagonal, square and triangular lattices. These results allow us to bound the ?$S$?-packing chromatic number for these graphs, with more precise bounds and exact values for sequences ?$S=(s_i,i \in \mathbb{N}^*)$?, …

Continuous energy-efficient monitoring model for mobile ad hoc networks

The monitoring of mobile ad hoc networks is an observation task that consists of analysing the operational status of these networks while evaluating their functionalities. In order to allow the whole network and applications to work properly, the monitoring task has become of considerable importance. It must be carried out in real-time by performing measurements, logs, configurations, etc. However, achieving continuous energy-efficient monitoring in mobile wireless networks is very challenging considering the environment features as well as the unpredictable behavior of the participating nodes. This paper outlines the challenges of continuous energy-efficient monitoring over mobile ad hoc n…

Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

Graphs and Algorithms

Grundy numbers of powers of graphs

International audience

A Graph Based Algorithm For Intersection Of Subdivision Surfaces

Computing surface intersections is a fundamental problem in geometric modeling. Any boolean operation can be seen as an intersection calculation followed by a selection of the parts necessary for building the surface of the resulting object. A robust and efficient algorithm to compute intersection on subdivision surfaces (surfaces generated by the Loop scheme) is proposed here. This algorithm relies on the concept of a bipartite graph which allows the reduction of the number of faces intersection tests. Intersection computations are accelerated by the use of the bipartite graph and the neighborhood of intersecting faces at a given level of subdivision to deduce intersecting faces at the fol…

On the family ofr-regular graphs with Grundy numberr+1

Abstract The Grundy number of a graph G , denoted by Γ ( G ) , is the largest k such that there exists a partition of V ( G ) , into k independent sets V 1 , … , V k and every vertex of V i is adjacent to at least one vertex in V j , for every j i . The objects which are studied in this article are families of r -regular graphs such that Γ ( G ) = r + 1 . Using the notion of independent module, a characterization of this family is given for r = 3 . Moreover, we determine classes of graphs in this family, in particular, the class of r -regular graphs without induced C 4 , for r ≤ 4 . Furthermore, our propositions imply results on the partial Grundy number.

Remarks on Partially Square Graphs, Hamiltonicity and Circumference