Given a finite or infinite graph G and positive integers `, h1, h2, h3, an L(h1, h2, h3)labelling of G with span ` is a mapping f : V (G) → {0, 1, 2, . . . , `} such that, for i = 1, 2, 3 and any u, v ∈ V (G) at distance i in G, |f(u)−f(v)| ≥ hi. A C(h1, h2, h3)-labelling of G with span ` is defined similarly by requiring |f(u)− f(v)|` ≥ hi instead, where |x|` = min{|x|, `− |x|}. The minimum sp...

Multi-atlas segmentation is a widely used tool in medical image analysis, providing robust and accurate results by learning from annotated atlas datasets. However, the availability of fully annotated atlas images for training is limited due to the time required for the labelling task. Segmentation methods requiring only a proportion of each atlas image to be labelled could therefore reduce the ...

Abstract E-super arithmetic graceful labelling of a graph G is bijection f from the union vertex set and edge to positive integers (1, 2, 3, … |V(G) ∪ E(G)|) such that edges have labels {1, …, |E(G)|} induced mapping * given by (uv) = f(u) + f(v) − f(uv) for uv ∈ E(G) has range {|V(G) E(G)| 1, |V(G)| 2|E(G)|} In this paper we prove H i (m, m) (1) chain even cycles C 4,n , 6,n are graceful.

In this paper some new type of odd graceful graphs are investigated. We prove that the graph obtained by joining a cycle C8 with some star graphs S1,r keeping one vertex and three vertices gap between pair of vertices of the cycle admits odd graceful labelling. We also prove that the graph obtained by joining a cycle C12 with some star graphs S1,r keeping two, three and five vertices gap betwee...

Kirousis, L.M., Fast parallel constraint satisfaction (Research Note), Artificial Intelligence 64 (1993) 147-160. We describe a class of constraint satisfaction problems (CSPs) for which a global solution can be found by a fast parallel algorithm. No relaxation preprocessing is needed for the parallel algorithm to work on this class of CSPs. The result is motivated from the problem of labelling...

Terrains are an essential part of outdoor environments. Terrain models are important for computer games and applications in architecture, urban design and archaeology. A popular and intuitive way to represent terrains is by contour maps. In order to render such representations in 3D the contours must be labelled with height values and converted to Digital Elevation Maps (DEM), which are regular...

We establish that all trees on at most 27 vertices admit graceful labellings and all trees on at most 26 vertices admit harmonious labellings. A graceful labelling of a graph G with q edges is an injection f : V (G) → {0, 1, 2, . . . , q} such that when each edge xy ∈ E(G) is assigned the label, |f(x) − f(y)|, all of the edge labels are distinct. A graph which admits a graceful labelling is sai...

Lecture notes by Matthew Dyer for lectures at the workshop “Coxeter groups and convex geometry.” 1. Root labelled Bruhat graph 1.1. Bruhat order. Let (W,S) be a Coxeter group. Bruhat order is the partial order defined by the following proposition; see [3] and [13] for more details. Proposition. There is a partial order ≤ on W with the following properties. Let w ∈ W and w = s1 . . . sn, n = l0(...

Dujmovi? et al. (FOCS2019) recently proved that every planar graph G is a subgraph of $$H\boxtimes P$$ , where $$\boxtimes$$ denotes the strong product, H treewidth 8 and P path. This result has found numerous applications to linear layouts, colouring, labelling. The proof given by based on similar decomposition Pilipczuk Siebertz (SODA2019) which constructive leads an $$O(n^2)$$ time algorithm...

The 1–2–3 Conjecture, raised by Karoński, Łuczak and Thomason, states that almost every graph G admits a proper 3-labelling, i.e., labelling of the edges with 1, 2, 3 such no two adjacent vertices are incident to same sum labels. Another interpretation this conjecture, may be attributed Chartrand et al., is can turned into locally irregular multigraph M, having degree, replacing each its at mos...