On Disjoint hypercubes in Fibonacci cubes

The {\em Fibonacci cube} of dimension $n$, denoted as $\Gamma\_n$, is the subgraph of $n$-cube $Q\_n$ induced by vertices with no consecutive 1's. We study the maximum number of disjoint subgraphs in $\Gamma\_n$ isomorphic to $Q\_k$, and denote this number by $q\_k(n)$. We prove several recursive results for $q\_k(n)$, in particular we prove that $q\_{k}(n) = q\_{k-1}(n-2) + q\_{k}(n-3)$. We also prove a closed formula in which $q\_k(n)$ is given in terms of Fibonacci numbers, and finally we give the generating function for the sequence $\{q\_{k}(n)\}\_{n=0}^{ \infty}$

Structural signatures of igneous sheet intrusion propagation

The geometry and distribution of planar igneous bodies (i.e. sheet intrusions), such as dykes, sills, and inclined sheets, has long been used to determine emplacement mechanics, define melt source locations, and reconstruct palaeostress conditions to shed light on various tectonic and magmatic processes. Since the 1970’s we have recognised that sheet intrusions do not necessarily display a continuous, planar geometry, but commonly consist of segments. The morphology of these segments and their connectors is controlled by, and provide insights into, the behaviour of the host rock during emplacement. For example, tensile brittle fracturing leads to the formation of intrusive steps or bridge structures between adjacent segments. In contrast, brittle shear faulting, cataclastic and ductile flow processes, as well as heat-induced viscous flow or fluidization, promotes magma finger development. Textural indicators of magma flow (e.g., rock fabrics) reveal that segments are aligned parallel to the initial sheet propagation direction. Recognising and mapping segment long axes thus allows melt source location hypotheses, derived from sheet distribution and orientation, to be robustly tested. Despite the information that can be obtained from these structural signatures of sheet intrusion propagation, they are largely overlooked by the structural and volcanological communities. To highlight their utility, we briefly review the formation of sheet intrusion segments, discuss how they inform interpretations of magma emplacement, and outline future research directions.Facultad de Ciencias Naturales y Muse

Edge-connectivity of strong products of graphs

The strong product G₁ ⊠ G₂ of graphs G₁ and G₂ is the graph with V(G₁)×V(G₂) as the vertex set, and two distinct vertices (x₁,x₂) and (y₁,y₂) are adjacent whenever for each i ∈ {1,2} either $x_i = y_i$ or $x_i y_i ∈ E(G_i)$. In this note we show that for two connected graphs G₁ and G₂ the edge-connectivity λ (G₁ ⊠ G₂) equals min{δ(G₁ ⊠ G₂), λ(G₁)(|V(G₂)| + 2|E(G₂)|), λ(G₂)(|V(G₁)| + 2|E(G₁)|)}. In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs