Partitioning Harary graphs into connected subgraphs containing prescribed vertices

International audienceA graph G is arbitrarily partitionable (AP for short) if for every partition (n_1, n_2, ..., n_p) of |V(G)| there exists a partition (V_1, V_2, ..., V_p) of V(G) such that each V_i induces a connected subgraph of G with order n_i. If, additionally, k of these subgraphs (k = 1 and n >= k

On locally irregular decompositions and the 1-2 Conjecture in digraphs

International audienceThe 1-2 Conjecture raised by Przybylo and Wozniak in 2010 asserts that every undirected graph admits a 2-total-weighting such that the sums of weights "incident" to the vertices yield a proper vertex-colouring. Following several recent works bringing related problems and notions (such as the well-known 1-2-3 Conjecture, and the notion of locally irregular decompositions) to digraphs, we here introduce and study several variants of the 1-2 Conjecture for digraphs. For every such variant, we raise conjectures concerning the number of weights necessary to obtain a desired total-weighting in any digraph. We verify some of these conjectures, while we obtain close results towards the ones that are still open

Neighbour-Sum-2-Distinguishing Edge-Weightings: Doubling the 1-2-3 Conjecture

International audienceThe 1-2-3 Conjecture asks whether every graph with no connected component isomorphic to K2 can be 3-edge-weighted so that every two adjacent vertices u and v can be distinguished via the sum of their incident weights, that is the incident sums of u and v differ by at least 1. We here investigate the consequences on the 1-2-3 Conjecture of requiring a stronger distinction condition. Namely, we consider two adjacent vertices distinguished when their incident sums differ by at least 2. As a guiding line, we conjecture that every graph with no connected component isomorphic to K2 admits a 5-edge-weighting permitting to distinguish the adjacent vertices in this stronger way. We verify this conjecture for several classes of graphs, including bipartite graphs and cubic graphs. We then consider algorithmic aspects, and show that it is NP-complete to determine the smallest k such that a given bipartite graph admits such a k-edge-weighting. In contrast, we show that the same problem can be solved in polynomial time for a given tree

Edge Weights and Vertex Colours: Minimizing Sum Count

International audienceNeighbour-sum-distinguishing edge-weightings are a way to "encode" proper vertex-colourings via the sums of weights incident to the vertices. Over the last decades, this notion has been attracting, in the context of several conjectures, ingrowing attention dedicated, notably, to understanding, which weights are needed to produce neighbour-sum-distinguishing edge-weightings for a given graph. This work is dedicated to investigating another related aspect, namely the minimum number of distinct sums/colours we can produce via a neighbour-sum-distinguishing edge-weighting of a given graph G, and the role of the assigned weights in that context. Clearly, this minimum number is bounded below by the chromatic number χ(G) of G. When using weights of ℤ, we show that, in general, we can produce neighbour-sum-distinguishing edge-weightings generating χ(G) distinct sums, except in the peculiar case where G is a balanced bipartite graph, in which case χ(G)+1 distinct sums can be generated. These results are best possible. When using k consecutive weights 1,...,k, we provide both lower and upper bounds, as a function of the maximum degree ∆, on the maximum least number of sums that can be generated for a graph with maximum degree ∆. For trees, which, in general, admit neighbour-sum-distinguishing 2-edge-weightings, we prove that this maximum, when using weights 1 and 2, is of order 2 log_2 ∆. Finally, we also establish the NP-hardness of several decision problems related to these questions

On locally irregular decompositions of subcubic graphs

International audienceA graph G is locally irregular if every two adjacent vertices of G have different degrees. A locally irregular decomposition of G is a partition E1,...,Ek of E(G) such that each G[Ei] is locally irregular. Not all graphs admit locally irregular decompositions, but for those who are decomposable, in that sense, it was conjectured by Baudon, Bensmail, Przybyło and Woźniak that they decompose into at most 3 locally irregular graphs. Towards that conjecture, it was recently proved by Bensmail, Merker and Thomassen that every decomposable graph decomposes into at most 328 locally irregular graphs.We here focus on locally irregular decompositions of subcubic graphs, which form an important family of graphs in this context, as all non-decomposable graphs are subcubic. As a main result, we prove that decomposable subcubic graphs decompose into at most 5 locally irregular graphs, and only at most 4 when the maximum average degree is less than 12/5. We then consider weaker decompositions, where subgraphs can also include regular connected components, and prove the relaxations of the conjecture above for subcubic graphs

A general decomposition theory for the 1-2-3 Conjecture and locally irregular decompositions

International audienceHow can one distinguish the adjacent vertices of a graph through an edge-weighting? In the last decades, this question has been attracting increasing attention, which resulted in the active field of distinguishing labellings. One of its most popular problems is the one where neighbours must be distinguishable via their incident sums of weights. An edge-weighting verifying this is said neighbour-sum-distinguishing. The popularity of this notion arises from two reasons. A first one is that designing a neighbour-sum-distinguishing edge-weighting showed up to be equivalent to turning a simple graph into a locally irregular (i.e., without neighbours with the same degree) multigraph by adding parallel edges, which is motivated by the concept of irregularity in graphs. Another source of popularity is probably the influence of the famous 1-2-3 Conjecture, which claims that such weightings with weights in {1,2,3} exist for graphs with no isolated edge. The 1-2-3 Conjecture has recently been investigated from a decompositional angle, via so-called locally irregular decompositions, which are edge-partitions into locally irregular subgraphs. Through several recent studies, it was shown that this concept is quite related to the 1-2-3 Conjecture. However, the full connexion between all those concepts was not clear. In this work, we propose an approach that generalizes all concepts above, involving coloured weights and sums. As a consequence, we get another interpretation of several existing results related to the 1-2-3 Conjecture. We also come up with new related conjectures, to which we give some support

Dépistage de la fibrose hépatique dans un service d'hépato-gastroentérologie chez des patients sans hépatopathie connue

ANGERS-BU Médecine-Pharmacie (490072105) / SudocPARIS-BIUM (751062103) / SudocSudocFranceF

Dépistage de la fibrose hépatique dans un service d'hépato-gastroentérologie chez des patients sans hépatopathie connue

ANGERS-BU Médecine-Pharmacie (490072105) / SudocPARIS-BIUM (751062103) / SudocSudocFranceF