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Onq-Power Cycles in Cubic Graphs

100%
Bensmail J.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 1
211-220
EN
In the context of a conjecture of Erdős and Gyárfás, we consider, for any q ≥ 2, the existence of q-power cycles (i.e., with length a power of q) in cubic graphs. We exhibit constructions showing that, for every q ≥ 3, there exist arbitrarily large cubic graphs with no q-power cycles. Concerning the remaining case q = 2 (which corresponds to the conjecture of Erdős and Gyárfás), we show that there exist arbitrarily large cubic graphs whose all 2-power cycles have length 4 only, or 8 only.
2
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Structural Properties of Recursively Partitionable Graphs with Connectivity 2

51%
Baudon O., Bensmail J., Foucaud F., Pilśniak M.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 1
89-115
EN
A connected graph G is said to be arbitrarily partitionable (AP for short) if for every partition (n1, . . . , np) of |V (G)| there exists a partition (V1, . . . , Vp) of V (G) such that each Vi induces a connected subgraph of G on ni vertices. Some stronger versions of this property were introduced, namely the ones of being online arbitrarily partitionable and recursively arbitrarily partitionable (OL-AP and R-AP for short, respectively), in which the subgraphs induced by a partition of G must not only be connected but also fulfil additional conditions. In this paper, we point out some structural properties of OL-AP and R-AP graphs with connectivity 2. In particular, we show that deleting a cut pair of these graphs results in a graph with a bounded number of components, some of whom have a small number of vertices. We obtain these results by studying a simple class of 2-connected graphs called balloons.
3
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An Oriented Version of the 1-2-3 Conjecture

51%
Baudon O., Bensmail J., Sopena É.
Discussiones Mathematicae Graph Theory
|
2015
|
tom 35
|
nr 1
141-156
EN
The well-known 1-2-3 Conjecture addressed by Karoński, Luczak and Thomason asks whether the edges of every undirected graph G with no isolated edge can be assigned weights from {1, 2, 3} so that the sum of incident weights at each vertex yields a proper vertex-colouring of G. In this work, we consider a similar problem for oriented graphs. We show that the arcs of every oriented graph −G⃗ can be assigned weights from {1, 2, 3} so that every two adjacent vertices of −G⃗ receive distinct sums of outgoing weights. This result is tight in the sense that some oriented graphs do not admit such an assignment using the weights from {1, 2} only. We finally prove that deciding whether two weights are sufficient for a given oriented graph is an NP-complete problem. These results also hold for product or list versions of this problem.
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